CHARACTERS OF HIGHEST WEIGHT MODULES AND

INTEGRABILITY

GURBIR DHILLON AND APOORVA KHARESTANFORD UNIVERSITY

Abstract We give positive formulas for the weights of every simple highest weight moduleL(λ) over an arbitrary KacndashMoody algebra Under a mild condition on the highest weightwe express the weights of L(λ) as an alternating sum similar to the WeylndashKac characterformula For general highest weight modules we answer questions of Bump and Lepowskyon weights and a question of Brion on the corresponding D-modules Many of these resultsare new even in finite type We prove similar assertions for highest weight modules over asymmetrizable quantum group

These results are obtained as elaborations of two ideas First the following data attachedto a highest weight module are equivalent (i) its integrability (ii) the convex hull of itsweights (iii) the Weyl group symmetry of its character and (iv) when a localization theoremis available its behavior on certain codimension one Schubert cells Second for special classesof highest weight modules including simple modules the above data determine the weights

Contents

1 Introduction 12 Statements of results 33 Preliminaries and notations 74 A classification of highest weight modules to first order 95 A question of Bump and weight formulas for non-integrable simple modules 126 A question of Lepowsky on the weights of highest weight modules 137 The WeylndashKac formula for the weights of simple modules 168 A question of Brion on localization of D-modules and a geometric interpretation

of integrability 189 Highest weight modules over symmetrizable quantum groups 21References 25

1 Introduction

Let g be a complex semisimple Lie algebra W its Weyl group and fix a triangular de-composition g = nminusoplus hoplus n+ The classical picture of the character of the finite-dimensionalsimple module of highest weight λ isin hlowast can be described as follows Qualitatively (i) theconvex hull of its weights is the polytope with vertices W (λ) and its weights are the inter-section of this with a suitable translate of the root lattice Equivalently (ii) the weights are

Date May 26 20172010 Mathematics Subject Classification Primary 17B10 Secondary 17B67 22E47 17B37 20G42Key words and phrases Highest weight module parabolic Verma module integrable Weyl group Tits

cone WeylndashKac formula weight formula BeilinsonndashBernstein localization quantum group

1

2 GURBIR DHILLON AND APOORVA KHARE

W invariant and those in the dominant chamber P+ are micro isin P+ micro 6 λ where 6 is theusual partial order on hlowast Quantitatively the exact multiplicities of the weights are given bythe comparatively sophisticated Weyl character formula [43]

When the simple highest weight module L(λ) is infinite-dimensional ie λ is no longerassumed to be dominant integral the character of L(λ) is known quantitatively throughKazhdanndashLusztig theory [4 10 27] For example when micro is a dominant integral weightw isinW we have in standard notation

chL(ww middot micro) =sumx6w

(minus1)`(w)minus`(x)Pxw(1) chM(xw middot micro) (11)

Just as in the special case of w = w ie the Weyl character formula due to the signs inEquation (11) it is hard to eyeball weight multiplicities and this problem is exacerbated bythe appearance of the KazhdanndashLusztig polynomials Pxw For this reason among others aproblem of longstanding interest is to obtain positive character formulae for L(λ)

With this motivation in this paper we provide such non-recursive formulae for the weightsof L(λ) ie determining for what micro isin hlowast the coefficient of emicro in (11) is nonzero In onedirection we provide descriptions of the weights of L(λ) analogous to (i) and (ii) above In asecond direction perhaps more surprisingly we obtain a version of the WeylndashKac characterformula for wtL(λ) ie involving only signs but no KazhdanndashLusztig polynomials

Moreover we obtain these results for g an arbitrary KacndashMoody algebra so they applyin particular in affine type to representation theory at the critical level for which charactersare not accessible by usual KazhdanndashLusztig theory but are of considerable interest eg inGeometric Langlands [1 14 15]

The aforementioned results emerge from a more general study of highest weight g-moduleswhose main ideas are distilled in the following theorem

Theorem 12 Let g be a KacndashMoody algebra with simple roots αi i isin I Fix λ isin hlowast andlet V be a g-module of highest weight λ The following data are equivalent

(1) IV the integrability of V ie IV = i isin I fi acts locally nilpotently on V (2) conv V the convex hull of the weights of V (3) The stabilizer of the character of V in W

Moreover if g is of finite type write Vrh for the perverse sheaf corresponding to V onthe flag variety and Xw for its supporting Schubert variety Then the data in (1)ndash(3) areequivalent to

(4) IVrh = i isin I the stalks of Vrh along the Schubert cell Csiw are nonzeroUnder this equivalence IV = IVrh Moreover for special classes of highest weight modulesincluding simple modules these data determine the weights

To our knowledge Theorem 12 is new for g of both finite and infinite type The datum ofthe theorem is an accessible lsquofirst-orderrsquo invariant yet the equivalence of its manifestations(1)ndash(4) has otherwise non-obvious consequences in the study of highest weight modulesThe usefulness of the theorem is that it allows one to pass freely between basic invariantsextracted from V via (1) restricting V to the copy of sl2 corresponding to each simpleroot (2) restricting V to a Cartan subalgebra h and using methods of convex geometry (3)restricting V to a Cartan subalgebra h and using the Weyl group action and (4) applyingBeilinsonndashBernstein localization and studying solutions of the D-module corresponding to V

In addition to the results for L(λ) alluded to above in this paper we apply Theorem 12to answer questions of Daniel Bump and James Lepowsky on the weights of highest weight

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 3

modules and a question of Michel Brion concerning localization of the corresponding D-modules on the flag variety Thus while our aims and inputs are classical Theorem 12seems to have resisted discovery until now and has nontrivial applications in highest weightrepresentation theory

Organization of the paper In Section 2 we describe our results and mention two interest-ing questions that may warrant further consideration After introducing notation in Section3 we prove in Section 4 the main result In Sections 5ndash8 we develop different applicationsof the main result Finally in Section 9 we discuss extensions of the previous sections toquantized enveloping algebras

2 Statements of results

Throughout the paper unless otherwise specified g is a KacndashMoody algebra over C withtriangular decomposition g = nminus oplus hoplus n+ and V is a g-module of highest weight λ isin hlowast

The ordering of this section is parallel to the organization of the paper and we refer thereader to Section 3 for notation

21 A classification of highest weight modules to first order We begin by provingthe equivalence of data (1)ndash(3) discussed in Theorem 12 which we repeat below for thereaderrsquos convenience Let I index the simple roots of g with corresponding simple loweringoperators fi i isin I

Theorem 21 The following data are equivalent

(1) IV the integrability of V ie IV = i isin I fi acts locally nilpotently on V (2) conv V the convex hull of the weights of V (3) The stabilizer of conv V in W the Weyl group

In particular the convex hull in (2) is always that of the parabolic Verma module M(λ IV )and the stabilizer in (3) is always the parabolic subgroup WIV

Let us indicate the sense in which the datum of Theorem 21 can be thought of as a lsquofirst-orderrsquo invariant of V Algebraically from (1) we see that this datum is sensitive only tothe action of each simple lowering operator fi on the highest weight line Categorically thisis the same as the JordanndashHolder content [V L(si middot λ)] for those i isin I with si middot λ 6 λGeometrically when localization is available the same si middot λ index certain Schubert divisorsin the support of V As we will see in Proposition 213 this datum records the genericbehaviour of V along these divisors and in particular is insensitive to any phenomenon incodimension at least two

In the proof of Theorem 21 the difficult implication is that the natural map M(λ IV )otimesVλ rarr V induces an equality on convex hulls of weights where Vλ is the highest weight lineof V To our knowledge this implication and in particular the theorem is new for g of bothfinite and infinite type It is somewhat remarkable that this theorem was not previouslyknown and we shall see it can be applied fruitfully to several expertsrsquo questions in highestweight representation theory

22 A question of Bump and weight formulas for non-integrable simple modulesIn this section we obtain three positive formulas for the weights of simple highest weightmodules wtL(λ) corresponding to the manifestations (1)ndash(3) of the datum of Theorem 12

The first formula uses restriction to the maximal standard Levi subalgebra whose actionis integrable

4 GURBIR DHILLON AND APOORVA KHARE

Proposition 22 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (23)

The second formula explicitly shows the relationship between wtL(λ) and its convex hull

Proposition 24wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (25)

The third formula uses the Weyl group action and a parabolic analogue of the dominantchamber introduced in Section 33

Proposition 26 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (27)

We remind that the assumption that λ has finite stabilizer is very mild as recalled in Propo-sition 32(4)

The question of whether Proposition 24 holds namely whether the weights of simplehighest weight modules are no finer an invariant than their hull was raised by Daniel BumpPropositions 22 24 and 26 are well known for integrable L(λ) and Propositions 22 and24 were proved by the second named author in finite type [28] The remaining cases inparticular Proposition 26 in all types are to our knowledge new

The obtained descriptions of wtL(λ) are particularly striking in infinite type When g isaffine the formulae are insensitive to whether λ is critical or non-critical In contrast criticallevel modules which figure prominently in the Geometric Langlands program [1 14 15]behave very differently from noncritical modules even at the level of characters When g issymmetrizable we similarly obtain weight formulae at critical λ The authors are unaware ofeven conjectural formulae for chL(λ) in this case Finally when g is non-symmetrizable itis unknown how to compute weight space multiplicities even for integrable L(λ) Thus for gnon-symmetrizable our formulae provide as much information on chL(λ) as one could hopefor given existing methods

We obtain the above three formulae using the following theorem For V a general highestweight module it gives a necessary and sufficient condition for M(λ IV ) rarr V to induce anequality of weights in terms of the action of a certain Levi subalgebra Define the potentialintegrability of V to be IpV = IL(λ) IV To justify the terminology note that these areprecisely the simple directions which become integrable in quotients of V

Theorem 28 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Theorem 28 is new in both finite and infinite type

23 A question of Lepowsky on the weights of highest weight modules In thissection we revisit the problem encountered in the proofs of Propositions 22 24 26 namelyunder what circumstances does the datum of Theorem 12 determine wtV

It was noticed by the second named author in [28] that wtV is a finer invariant thanconv V whenever the potential integrability IpV contains orthogonal roots The prototypicalexample here is for g = sl2 oplus sl2 and V = M(0)M(s1s2 middot 0)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 5

James Lepowsky asked whether this orthogonality is the only obstruction to determiningwtV from its convex hull In finite type this would reduce to a question in types A2 B2and G2 We now answer this question affirmatively for g an arbitrary KacndashMoody algebra

Theorem 29 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if IL(λ) J is complete ie (αi αj) 6=0 foralli j isin IL(λ) J

We are not aware of a precursor to Theorem 29 in the literature even in finite type

24 The WeylndashKac formula for the weights of simple modules We prove the fol-lowing identity whose notation we will explain below

Theorem 210 For λ isin hlowast such that the stabilizer of λ in WIL(λ)is finite we have

wtL(λ) =sum

wisinWIL(λ)

weλprod

iisinI(1minus eminusαi) (211)

On the left hand side of (211) we mean the lsquomultiplicity-freersquo charactersum

microisinhlowastL(λ)micro 6=0 emicro

On the right hand side of (211) in each summand we take the lsquohighest weightrsquo expansion ofw(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0

In particular as claimed in the introduction each summand comes with signs but no multi-plicities as in KazhdanndashLusztig theory

Precursors to Theorem 210 include Kass [26] Brion [8] Walton [42] Postnikov [39] andSchutzer [40] for the integrable highest weight modules As pointed out to us by MichelBrion in finite type Theorem 210 follows from a more general formula for exponential sumsover polyhedra cf Remark 79 below All other cases are to our knowledge new

25 A question of Brion on localization of D-modules and a geometric interpre-tation of integrability In this section we use a topological manifestation of the datum inTheorem 12 to answer a question of Brion in geometric representation theory

Brionrsquos question is as follows Let g be of finite type λ a regular dominant integral weightand consider convL(λ) the Weyl polytope For any J sub I if one intersects the tangentcones of convL(λ) at wλw isin WJ one obtains convM(λ J) Brion first asked whetheran analogous formula holds for other highest weight modules We answer this questionaffirmatively in [11]

Brion further observed that this procedure of localizing convL(λ) in convex geometry isthe shadow of a localization in complex geometry More precisely let X denote the flagvariety and Lλ denote the line bundle on X with H0(XLλ) L(λ) As usual write Xw

for the closure of the Bruhat cell Cw = BwBB forallw isin W and write w for the longestelement of W Let D denote the standard dualities on Category O and regular holonomicD-modules Then for a union of Schubert divisors Z = cupiisinIJXsiw with complement U

we have H0(ULλ) DM(λ J) Thus in this case localization of the convex hull could berecovered as taking the convex hull of the weights of sections of Lλ on an appropriate openset U Brion asked whether a similar result holds for more general highest weight modules

We answer this affirmatively for a regular integral infinitesimal character By translationit suffices to examine the regular block O0

6 GURBIR DHILLON AND APOORVA KHARE

Theorem 212 Let g be of finite type and λ = wmiddotminus2ρ Let V be a g-module of highest weightλ and write V for the corresponding D-module on X For J sub IV set Z = cupiisinIV JXsiw

and write GB = Z t U Then we have DH0(U DV) is a g-module of highest weight λ andintegrability J

Let us mention one ingredient in the proof of Theorem 212 Let Vrh denote the perversesheaf corresponding to V under the RiemannndashHilbert correspondence Then the datum ofTheorem 12 manifests as

Proposition 213 For J sub IL(λ) consider the smooth open subvariety of Xw given oncomplex points by

UJ = Cw t⊔iisinJ

Csiw (214)

Letting VVrh be as above upon restricting to UIL(λ)we have

Vrh|UIL(λ) jCUIV [`(w)] (215)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

To our knowledge Proposition 213 is new though not difficult to prove once stated Theproof is a pleasing geometric avatar of the idea that M(λ IV ) rarr V is an isomorphism lsquotofirst orderrsquo

26 Highest weight modules over symmetrizable quantum groups In the final sec-tion we apply both the methods and the results from earlier to study highest weight modulesover quantum groups and obtain results similar to those discussed above

27 Further problems We conclude this section by calling attention to two problemssuggested by our results

Problem 1 WeylndashKac formula with infinite stabilizers The statement of Theorem 210includes the assumption that the highest weight has finite integrable stabilizer However theasserted identity otherwise tends to fail in interesting ways For example

Proposition 216 For g of rank 2 and the trivial module L(0) we havesumwisinW

w1

(1minus eminusα1)(1minus eminusα2)= 1 +

sumαisin∆im

minus

eα (217)

where ∆imminus denotes the set of negative imaginary roots

It would be interesting to formulate a correct version of Theorem 210 even for one-dimensionalmodules in affine type which might be thought of as a multiplicity-free Macdonald identityIndeed Equation (217) suggests correction terms coming from imaginary roots akin to [24]

Problem 2 From convex hulls to weights The regular block of Category O is the prototypi-cal example of a category of infinite-dimensional g-modules of perverse sheaves on a stratifiedspace and of a highest weight category Accordingly it has very rich representation-theoreticgeometric and purely categorical structure The main theorem of the paper Theorem 12determines the meaning of the convex hull of the weights in these three guises namely theintegrability behavior in codimension one and the JordanndashHolder content [V L(si middot λ)] Itwould be very interesting to obtain a similar understanding of the weights themselves

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7

Acknowledgments It is a pleasure to thank Michel Brion Daniel Bump Shrawan KumarJames Lepowsky Sam Raskin Geordie Williamson and Zhiwei Yun for valuable discussionsThe work of GD was partially supported by the Department of Defense (DoD) through theNDSEG fellowship

3 Preliminaries and notations

The contents of this section are mostly standard We advise the reader to skim Subsection33 and refer back to the rest only as needed

31 Notation for numbers and sums We write Z for the integers and QRC for therational real and complex numbers respectively For a subset S of a real vector space E wewrite Zgt0S for the set of finite linear combinations of S with coefficients in Zgt0 and similarlyZSQgt0SRS etc

32 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [25] and [33] In this paper we work throughout over C Let I bea finite set and A = (aij)ijisinI a generalized Cartan matrix Fix a realization (h π π) withsimple roots π = αiiisinI sub hlowast and coroots π = αiiisinI sub h satisfying (αi αj) = aij foralli j isin I

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h αi)ei [h fi] = minus(h αi)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially thesecoincide when A is symmetrizable When A is clear from context we will abbreviate theseto g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 forα isin ∆+ and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0

write suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin Jand h For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λ Writing AJfor the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJ as a

subalgebra of g(A) Now write πJ ∆+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finallywe define the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

2 GURBIR DHILLON AND APOORVA KHARE

W invariant and those in the dominant chamber P+ are micro isin P+ micro 6 λ where 6 is theusual partial order on hlowast Quantitatively the exact multiplicities of the weights are given bythe comparatively sophisticated Weyl character formula [43]

When the simple highest weight module L(λ) is infinite-dimensional ie λ is no longerassumed to be dominant integral the character of L(λ) is known quantitatively throughKazhdanndashLusztig theory [4 10 27] For example when micro is a dominant integral weightw isinW we have in standard notation

chL(ww middot micro) =sumx6w

(minus1)`(w)minus`(x)Pxw(1) chM(xw middot micro) (11)

Just as in the special case of w = w ie the Weyl character formula due to the signs inEquation (11) it is hard to eyeball weight multiplicities and this problem is exacerbated bythe appearance of the KazhdanndashLusztig polynomials Pxw For this reason among others aproblem of longstanding interest is to obtain positive character formulae for L(λ)

With this motivation in this paper we provide such non-recursive formulae for the weightsof L(λ) ie determining for what micro isin hlowast the coefficient of emicro in (11) is nonzero In onedirection we provide descriptions of the weights of L(λ) analogous to (i) and (ii) above In asecond direction perhaps more surprisingly we obtain a version of the WeylndashKac characterformula for wtL(λ) ie involving only signs but no KazhdanndashLusztig polynomials

Moreover we obtain these results for g an arbitrary KacndashMoody algebra so they applyin particular in affine type to representation theory at the critical level for which charactersare not accessible by usual KazhdanndashLusztig theory but are of considerable interest eg inGeometric Langlands [1 14 15]

The aforementioned results emerge from a more general study of highest weight g-moduleswhose main ideas are distilled in the following theorem

Theorem 12 Let g be a KacndashMoody algebra with simple roots αi i isin I Fix λ isin hlowast andlet V be a g-module of highest weight λ The following data are equivalent

(1) IV the integrability of V ie IV = i isin I fi acts locally nilpotently on V (2) conv V the convex hull of the weights of V (3) The stabilizer of the character of V in W

Moreover if g is of finite type write Vrh for the perverse sheaf corresponding to V onthe flag variety and Xw for its supporting Schubert variety Then the data in (1)ndash(3) areequivalent to

(4) IVrh = i isin I the stalks of Vrh along the Schubert cell Csiw are nonzeroUnder this equivalence IV = IVrh Moreover for special classes of highest weight modulesincluding simple modules these data determine the weights

To our knowledge Theorem 12 is new for g of both finite and infinite type The datum ofthe theorem is an accessible lsquofirst-orderrsquo invariant yet the equivalence of its manifestations(1)ndash(4) has otherwise non-obvious consequences in the study of highest weight modulesThe usefulness of the theorem is that it allows one to pass freely between basic invariantsextracted from V via (1) restricting V to the copy of sl2 corresponding to each simpleroot (2) restricting V to a Cartan subalgebra h and using methods of convex geometry (3)restricting V to a Cartan subalgebra h and using the Weyl group action and (4) applyingBeilinsonndashBernstein localization and studying solutions of the D-module corresponding to V

In addition to the results for L(λ) alluded to above in this paper we apply Theorem 12to answer questions of Daniel Bump and James Lepowsky on the weights of highest weight

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 3

modules and a question of Michel Brion concerning localization of the corresponding D-modules on the flag variety Thus while our aims and inputs are classical Theorem 12seems to have resisted discovery until now and has nontrivial applications in highest weightrepresentation theory

Organization of the paper In Section 2 we describe our results and mention two interest-ing questions that may warrant further consideration After introducing notation in Section3 we prove in Section 4 the main result In Sections 5ndash8 we develop different applicationsof the main result Finally in Section 9 we discuss extensions of the previous sections toquantized enveloping algebras

2 Statements of results

Throughout the paper unless otherwise specified g is a KacndashMoody algebra over C withtriangular decomposition g = nminus oplus hoplus n+ and V is a g-module of highest weight λ isin hlowast

The ordering of this section is parallel to the organization of the paper and we refer thereader to Section 3 for notation

21 A classification of highest weight modules to first order We begin by provingthe equivalence of data (1)ndash(3) discussed in Theorem 12 which we repeat below for thereaderrsquos convenience Let I index the simple roots of g with corresponding simple loweringoperators fi i isin I

Theorem 21 The following data are equivalent

(1) IV the integrability of V ie IV = i isin I fi acts locally nilpotently on V (2) conv V the convex hull of the weights of V (3) The stabilizer of conv V in W the Weyl group

In particular the convex hull in (2) is always that of the parabolic Verma module M(λ IV )and the stabilizer in (3) is always the parabolic subgroup WIV

Let us indicate the sense in which the datum of Theorem 21 can be thought of as a lsquofirst-orderrsquo invariant of V Algebraically from (1) we see that this datum is sensitive only tothe action of each simple lowering operator fi on the highest weight line Categorically thisis the same as the JordanndashHolder content [V L(si middot λ)] for those i isin I with si middot λ 6 λGeometrically when localization is available the same si middot λ index certain Schubert divisorsin the support of V As we will see in Proposition 213 this datum records the genericbehaviour of V along these divisors and in particular is insensitive to any phenomenon incodimension at least two

In the proof of Theorem 21 the difficult implication is that the natural map M(λ IV )otimesVλ rarr V induces an equality on convex hulls of weights where Vλ is the highest weight lineof V To our knowledge this implication and in particular the theorem is new for g of bothfinite and infinite type It is somewhat remarkable that this theorem was not previouslyknown and we shall see it can be applied fruitfully to several expertsrsquo questions in highestweight representation theory

22 A question of Bump and weight formulas for non-integrable simple modulesIn this section we obtain three positive formulas for the weights of simple highest weightmodules wtL(λ) corresponding to the manifestations (1)ndash(3) of the datum of Theorem 12

The first formula uses restriction to the maximal standard Levi subalgebra whose actionis integrable

4 GURBIR DHILLON AND APOORVA KHARE

Proposition 22 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (23)

The second formula explicitly shows the relationship between wtL(λ) and its convex hull

Proposition 24wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (25)

The third formula uses the Weyl group action and a parabolic analogue of the dominantchamber introduced in Section 33

Proposition 26 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (27)

We remind that the assumption that λ has finite stabilizer is very mild as recalled in Propo-sition 32(4)

The question of whether Proposition 24 holds namely whether the weights of simplehighest weight modules are no finer an invariant than their hull was raised by Daniel BumpPropositions 22 24 and 26 are well known for integrable L(λ) and Propositions 22 and24 were proved by the second named author in finite type [28] The remaining cases inparticular Proposition 26 in all types are to our knowledge new

The obtained descriptions of wtL(λ) are particularly striking in infinite type When g isaffine the formulae are insensitive to whether λ is critical or non-critical In contrast criticallevel modules which figure prominently in the Geometric Langlands program [1 14 15]behave very differently from noncritical modules even at the level of characters When g issymmetrizable we similarly obtain weight formulae at critical λ The authors are unaware ofeven conjectural formulae for chL(λ) in this case Finally when g is non-symmetrizable itis unknown how to compute weight space multiplicities even for integrable L(λ) Thus for gnon-symmetrizable our formulae provide as much information on chL(λ) as one could hopefor given existing methods

We obtain the above three formulae using the following theorem For V a general highestweight module it gives a necessary and sufficient condition for M(λ IV ) rarr V to induce anequality of weights in terms of the action of a certain Levi subalgebra Define the potentialintegrability of V to be IpV = IL(λ) IV To justify the terminology note that these areprecisely the simple directions which become integrable in quotients of V

Theorem 28 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Theorem 28 is new in both finite and infinite type

23 A question of Lepowsky on the weights of highest weight modules In thissection we revisit the problem encountered in the proofs of Propositions 22 24 26 namelyunder what circumstances does the datum of Theorem 12 determine wtV

It was noticed by the second named author in [28] that wtV is a finer invariant thanconv V whenever the potential integrability IpV contains orthogonal roots The prototypicalexample here is for g = sl2 oplus sl2 and V = M(0)M(s1s2 middot 0)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 5

James Lepowsky asked whether this orthogonality is the only obstruction to determiningwtV from its convex hull In finite type this would reduce to a question in types A2 B2and G2 We now answer this question affirmatively for g an arbitrary KacndashMoody algebra

Theorem 29 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if IL(λ) J is complete ie (αi αj) 6=0 foralli j isin IL(λ) J

We are not aware of a precursor to Theorem 29 in the literature even in finite type

24 The WeylndashKac formula for the weights of simple modules We prove the fol-lowing identity whose notation we will explain below

Theorem 210 For λ isin hlowast such that the stabilizer of λ in WIL(λ)is finite we have

wtL(λ) =sum

wisinWIL(λ)

weλprod

iisinI(1minus eminusαi) (211)

On the left hand side of (211) we mean the lsquomultiplicity-freersquo charactersum

microisinhlowastL(λ)micro 6=0 emicro

On the right hand side of (211) in each summand we take the lsquohighest weightrsquo expansion ofw(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0

In particular as claimed in the introduction each summand comes with signs but no multi-plicities as in KazhdanndashLusztig theory

Precursors to Theorem 210 include Kass [26] Brion [8] Walton [42] Postnikov [39] andSchutzer [40] for the integrable highest weight modules As pointed out to us by MichelBrion in finite type Theorem 210 follows from a more general formula for exponential sumsover polyhedra cf Remark 79 below All other cases are to our knowledge new

25 A question of Brion on localization of D-modules and a geometric interpre-tation of integrability In this section we use a topological manifestation of the datum inTheorem 12 to answer a question of Brion in geometric representation theory

Brionrsquos question is as follows Let g be of finite type λ a regular dominant integral weightand consider convL(λ) the Weyl polytope For any J sub I if one intersects the tangentcones of convL(λ) at wλw isin WJ one obtains convM(λ J) Brion first asked whetheran analogous formula holds for other highest weight modules We answer this questionaffirmatively in [11]

Brion further observed that this procedure of localizing convL(λ) in convex geometry isthe shadow of a localization in complex geometry More precisely let X denote the flagvariety and Lλ denote the line bundle on X with H0(XLλ) L(λ) As usual write Xw

for the closure of the Bruhat cell Cw = BwBB forallw isin W and write w for the longestelement of W Let D denote the standard dualities on Category O and regular holonomicD-modules Then for a union of Schubert divisors Z = cupiisinIJXsiw with complement U

we have H0(ULλ) DM(λ J) Thus in this case localization of the convex hull could berecovered as taking the convex hull of the weights of sections of Lλ on an appropriate openset U Brion asked whether a similar result holds for more general highest weight modules

We answer this affirmatively for a regular integral infinitesimal character By translationit suffices to examine the regular block O0

6 GURBIR DHILLON AND APOORVA KHARE

Theorem 212 Let g be of finite type and λ = wmiddotminus2ρ Let V be a g-module of highest weightλ and write V for the corresponding D-module on X For J sub IV set Z = cupiisinIV JXsiw

and write GB = Z t U Then we have DH0(U DV) is a g-module of highest weight λ andintegrability J

Let us mention one ingredient in the proof of Theorem 212 Let Vrh denote the perversesheaf corresponding to V under the RiemannndashHilbert correspondence Then the datum ofTheorem 12 manifests as

Proposition 213 For J sub IL(λ) consider the smooth open subvariety of Xw given oncomplex points by

UJ = Cw t⊔iisinJ

Csiw (214)

Letting VVrh be as above upon restricting to UIL(λ)we have

Vrh|UIL(λ) jCUIV [`(w)] (215)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

To our knowledge Proposition 213 is new though not difficult to prove once stated Theproof is a pleasing geometric avatar of the idea that M(λ IV ) rarr V is an isomorphism lsquotofirst orderrsquo

26 Highest weight modules over symmetrizable quantum groups In the final sec-tion we apply both the methods and the results from earlier to study highest weight modulesover quantum groups and obtain results similar to those discussed above

27 Further problems We conclude this section by calling attention to two problemssuggested by our results

Problem 1 WeylndashKac formula with infinite stabilizers The statement of Theorem 210includes the assumption that the highest weight has finite integrable stabilizer However theasserted identity otherwise tends to fail in interesting ways For example

Proposition 216 For g of rank 2 and the trivial module L(0) we havesumwisinW

w1

(1minus eminusα1)(1minus eminusα2)= 1 +

sumαisin∆im

minus

eα (217)

where ∆imminus denotes the set of negative imaginary roots

It would be interesting to formulate a correct version of Theorem 210 even for one-dimensionalmodules in affine type which might be thought of as a multiplicity-free Macdonald identityIndeed Equation (217) suggests correction terms coming from imaginary roots akin to [24]

Problem 2 From convex hulls to weights The regular block of Category O is the prototypi-cal example of a category of infinite-dimensional g-modules of perverse sheaves on a stratifiedspace and of a highest weight category Accordingly it has very rich representation-theoreticgeometric and purely categorical structure The main theorem of the paper Theorem 12determines the meaning of the convex hull of the weights in these three guises namely theintegrability behavior in codimension one and the JordanndashHolder content [V L(si middot λ)] Itwould be very interesting to obtain a similar understanding of the weights themselves

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7

Acknowledgments It is a pleasure to thank Michel Brion Daniel Bump Shrawan KumarJames Lepowsky Sam Raskin Geordie Williamson and Zhiwei Yun for valuable discussionsThe work of GD was partially supported by the Department of Defense (DoD) through theNDSEG fellowship

3 Preliminaries and notations

The contents of this section are mostly standard We advise the reader to skim Subsection33 and refer back to the rest only as needed

31 Notation for numbers and sums We write Z for the integers and QRC for therational real and complex numbers respectively For a subset S of a real vector space E wewrite Zgt0S for the set of finite linear combinations of S with coefficients in Zgt0 and similarlyZSQgt0SRS etc

32 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [25] and [33] In this paper we work throughout over C Let I bea finite set and A = (aij)ijisinI a generalized Cartan matrix Fix a realization (h π π) withsimple roots π = αiiisinI sub hlowast and coroots π = αiiisinI sub h satisfying (αi αj) = aij foralli j isin I

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h αi)ei [h fi] = minus(h αi)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially thesecoincide when A is symmetrizable When A is clear from context we will abbreviate theseto g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 forα isin ∆+ and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0

write suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin Jand h For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λ Writing AJfor the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJ as a

subalgebra of g(A) Now write πJ ∆+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finallywe define the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 3

modules and a question of Michel Brion concerning localization of the corresponding D-modules on the flag variety Thus while our aims and inputs are classical Theorem 12seems to have resisted discovery until now and has nontrivial applications in highest weightrepresentation theory

Organization of the paper In Section 2 we describe our results and mention two interest-ing questions that may warrant further consideration After introducing notation in Section3 we prove in Section 4 the main result In Sections 5ndash8 we develop different applicationsof the main result Finally in Section 9 we discuss extensions of the previous sections toquantized enveloping algebras

2 Statements of results

Throughout the paper unless otherwise specified g is a KacndashMoody algebra over C withtriangular decomposition g = nminus oplus hoplus n+ and V is a g-module of highest weight λ isin hlowast

The ordering of this section is parallel to the organization of the paper and we refer thereader to Section 3 for notation

21 A classification of highest weight modules to first order We begin by provingthe equivalence of data (1)ndash(3) discussed in Theorem 12 which we repeat below for thereaderrsquos convenience Let I index the simple roots of g with corresponding simple loweringoperators fi i isin I

Theorem 21 The following data are equivalent

(1) IV the integrability of V ie IV = i isin I fi acts locally nilpotently on V (2) conv V the convex hull of the weights of V (3) The stabilizer of conv V in W the Weyl group

In particular the convex hull in (2) is always that of the parabolic Verma module M(λ IV )and the stabilizer in (3) is always the parabolic subgroup WIV

Let us indicate the sense in which the datum of Theorem 21 can be thought of as a lsquofirst-orderrsquo invariant of V Algebraically from (1) we see that this datum is sensitive only tothe action of each simple lowering operator fi on the highest weight line Categorically thisis the same as the JordanndashHolder content [V L(si middot λ)] for those i isin I with si middot λ 6 λGeometrically when localization is available the same si middot λ index certain Schubert divisorsin the support of V As we will see in Proposition 213 this datum records the genericbehaviour of V along these divisors and in particular is insensitive to any phenomenon incodimension at least two

In the proof of Theorem 21 the difficult implication is that the natural map M(λ IV )otimesVλ rarr V induces an equality on convex hulls of weights where Vλ is the highest weight lineof V To our knowledge this implication and in particular the theorem is new for g of bothfinite and infinite type It is somewhat remarkable that this theorem was not previouslyknown and we shall see it can be applied fruitfully to several expertsrsquo questions in highestweight representation theory

22 A question of Bump and weight formulas for non-integrable simple modulesIn this section we obtain three positive formulas for the weights of simple highest weightmodules wtL(λ) corresponding to the manifestations (1)ndash(3) of the datum of Theorem 12

The first formula uses restriction to the maximal standard Levi subalgebra whose actionis integrable

4 GURBIR DHILLON AND APOORVA KHARE

Proposition 22 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (23)

The second formula explicitly shows the relationship between wtL(λ) and its convex hull

Proposition 24wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (25)

The third formula uses the Weyl group action and a parabolic analogue of the dominantchamber introduced in Section 33

Proposition 26 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (27)

We remind that the assumption that λ has finite stabilizer is very mild as recalled in Propo-sition 32(4)

The question of whether Proposition 24 holds namely whether the weights of simplehighest weight modules are no finer an invariant than their hull was raised by Daniel BumpPropositions 22 24 and 26 are well known for integrable L(λ) and Propositions 22 and24 were proved by the second named author in finite type [28] The remaining cases inparticular Proposition 26 in all types are to our knowledge new

The obtained descriptions of wtL(λ) are particularly striking in infinite type When g isaffine the formulae are insensitive to whether λ is critical or non-critical In contrast criticallevel modules which figure prominently in the Geometric Langlands program [1 14 15]behave very differently from noncritical modules even at the level of characters When g issymmetrizable we similarly obtain weight formulae at critical λ The authors are unaware ofeven conjectural formulae for chL(λ) in this case Finally when g is non-symmetrizable itis unknown how to compute weight space multiplicities even for integrable L(λ) Thus for gnon-symmetrizable our formulae provide as much information on chL(λ) as one could hopefor given existing methods

We obtain the above three formulae using the following theorem For V a general highestweight module it gives a necessary and sufficient condition for M(λ IV ) rarr V to induce anequality of weights in terms of the action of a certain Levi subalgebra Define the potentialintegrability of V to be IpV = IL(λ) IV To justify the terminology note that these areprecisely the simple directions which become integrable in quotients of V

Theorem 28 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Theorem 28 is new in both finite and infinite type

23 A question of Lepowsky on the weights of highest weight modules In thissection we revisit the problem encountered in the proofs of Propositions 22 24 26 namelyunder what circumstances does the datum of Theorem 12 determine wtV

It was noticed by the second named author in [28] that wtV is a finer invariant thanconv V whenever the potential integrability IpV contains orthogonal roots The prototypicalexample here is for g = sl2 oplus sl2 and V = M(0)M(s1s2 middot 0)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 5

James Lepowsky asked whether this orthogonality is the only obstruction to determiningwtV from its convex hull In finite type this would reduce to a question in types A2 B2and G2 We now answer this question affirmatively for g an arbitrary KacndashMoody algebra

Theorem 29 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if IL(λ) J is complete ie (αi αj) 6=0 foralli j isin IL(λ) J

We are not aware of a precursor to Theorem 29 in the literature even in finite type

24 The WeylndashKac formula for the weights of simple modules We prove the fol-lowing identity whose notation we will explain below

Theorem 210 For λ isin hlowast such that the stabilizer of λ in WIL(λ)is finite we have

wtL(λ) =sum

wisinWIL(λ)

weλprod

iisinI(1minus eminusαi) (211)

On the left hand side of (211) we mean the lsquomultiplicity-freersquo charactersum

microisinhlowastL(λ)micro 6=0 emicro

On the right hand side of (211) in each summand we take the lsquohighest weightrsquo expansion ofw(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0

In particular as claimed in the introduction each summand comes with signs but no multi-plicities as in KazhdanndashLusztig theory

Precursors to Theorem 210 include Kass [26] Brion [8] Walton [42] Postnikov [39] andSchutzer [40] for the integrable highest weight modules As pointed out to us by MichelBrion in finite type Theorem 210 follows from a more general formula for exponential sumsover polyhedra cf Remark 79 below All other cases are to our knowledge new

25 A question of Brion on localization of D-modules and a geometric interpre-tation of integrability In this section we use a topological manifestation of the datum inTheorem 12 to answer a question of Brion in geometric representation theory

Brionrsquos question is as follows Let g be of finite type λ a regular dominant integral weightand consider convL(λ) the Weyl polytope For any J sub I if one intersects the tangentcones of convL(λ) at wλw isin WJ one obtains convM(λ J) Brion first asked whetheran analogous formula holds for other highest weight modules We answer this questionaffirmatively in [11]

Brion further observed that this procedure of localizing convL(λ) in convex geometry isthe shadow of a localization in complex geometry More precisely let X denote the flagvariety and Lλ denote the line bundle on X with H0(XLλ) L(λ) As usual write Xw

for the closure of the Bruhat cell Cw = BwBB forallw isin W and write w for the longestelement of W Let D denote the standard dualities on Category O and regular holonomicD-modules Then for a union of Schubert divisors Z = cupiisinIJXsiw with complement U

we have H0(ULλ) DM(λ J) Thus in this case localization of the convex hull could berecovered as taking the convex hull of the weights of sections of Lλ on an appropriate openset U Brion asked whether a similar result holds for more general highest weight modules

We answer this affirmatively for a regular integral infinitesimal character By translationit suffices to examine the regular block O0

6 GURBIR DHILLON AND APOORVA KHARE

Theorem 212 Let g be of finite type and λ = wmiddotminus2ρ Let V be a g-module of highest weightλ and write V for the corresponding D-module on X For J sub IV set Z = cupiisinIV JXsiw

and write GB = Z t U Then we have DH0(U DV) is a g-module of highest weight λ andintegrability J

Let us mention one ingredient in the proof of Theorem 212 Let Vrh denote the perversesheaf corresponding to V under the RiemannndashHilbert correspondence Then the datum ofTheorem 12 manifests as

Proposition 213 For J sub IL(λ) consider the smooth open subvariety of Xw given oncomplex points by

UJ = Cw t⊔iisinJ

Csiw (214)

Letting VVrh be as above upon restricting to UIL(λ)we have

Vrh|UIL(λ) jCUIV [`(w)] (215)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

To our knowledge Proposition 213 is new though not difficult to prove once stated Theproof is a pleasing geometric avatar of the idea that M(λ IV ) rarr V is an isomorphism lsquotofirst orderrsquo

26 Highest weight modules over symmetrizable quantum groups In the final sec-tion we apply both the methods and the results from earlier to study highest weight modulesover quantum groups and obtain results similar to those discussed above

27 Further problems We conclude this section by calling attention to two problemssuggested by our results

Problem 1 WeylndashKac formula with infinite stabilizers The statement of Theorem 210includes the assumption that the highest weight has finite integrable stabilizer However theasserted identity otherwise tends to fail in interesting ways For example

Proposition 216 For g of rank 2 and the trivial module L(0) we havesumwisinW

w1

(1minus eminusα1)(1minus eminusα2)= 1 +

sumαisin∆im

minus

eα (217)

where ∆imminus denotes the set of negative imaginary roots

It would be interesting to formulate a correct version of Theorem 210 even for one-dimensionalmodules in affine type which might be thought of as a multiplicity-free Macdonald identityIndeed Equation (217) suggests correction terms coming from imaginary roots akin to [24]

Problem 2 From convex hulls to weights The regular block of Category O is the prototypi-cal example of a category of infinite-dimensional g-modules of perverse sheaves on a stratifiedspace and of a highest weight category Accordingly it has very rich representation-theoreticgeometric and purely categorical structure The main theorem of the paper Theorem 12determines the meaning of the convex hull of the weights in these three guises namely theintegrability behavior in codimension one and the JordanndashHolder content [V L(si middot λ)] Itwould be very interesting to obtain a similar understanding of the weights themselves

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7

Acknowledgments It is a pleasure to thank Michel Brion Daniel Bump Shrawan KumarJames Lepowsky Sam Raskin Geordie Williamson and Zhiwei Yun for valuable discussionsThe work of GD was partially supported by the Department of Defense (DoD) through theNDSEG fellowship

3 Preliminaries and notations

The contents of this section are mostly standard We advise the reader to skim Subsection33 and refer back to the rest only as needed

31 Notation for numbers and sums We write Z for the integers and QRC for therational real and complex numbers respectively For a subset S of a real vector space E wewrite Zgt0S for the set of finite linear combinations of S with coefficients in Zgt0 and similarlyZSQgt0SRS etc

32 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [25] and [33] In this paper we work throughout over C Let I bea finite set and A = (aij)ijisinI a generalized Cartan matrix Fix a realization (h π π) withsimple roots π = αiiisinI sub hlowast and coroots π = αiiisinI sub h satisfying (αi αj) = aij foralli j isin I

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h αi)ei [h fi] = minus(h αi)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially thesecoincide when A is symmetrizable When A is clear from context we will abbreviate theseto g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 forα isin ∆+ and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0

write suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin Jand h For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λ Writing AJfor the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJ as a

subalgebra of g(A) Now write πJ ∆+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finallywe define the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

4 GURBIR DHILLON AND APOORVA KHARE

Proposition 22 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (23)

The second formula explicitly shows the relationship between wtL(λ) and its convex hull

Proposition 24wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (25)

The third formula uses the Weyl group action and a parabolic analogue of the dominantchamber introduced in Section 33

Proposition 26 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (27)

We remind that the assumption that λ has finite stabilizer is very mild as recalled in Propo-sition 32(4)

The question of whether Proposition 24 holds namely whether the weights of simplehighest weight modules are no finer an invariant than their hull was raised by Daniel BumpPropositions 22 24 and 26 are well known for integrable L(λ) and Propositions 22 and24 were proved by the second named author in finite type [28] The remaining cases inparticular Proposition 26 in all types are to our knowledge new

The obtained descriptions of wtL(λ) are particularly striking in infinite type When g isaffine the formulae are insensitive to whether λ is critical or non-critical In contrast criticallevel modules which figure prominently in the Geometric Langlands program [1 14 15]behave very differently from noncritical modules even at the level of characters When g issymmetrizable we similarly obtain weight formulae at critical λ The authors are unaware ofeven conjectural formulae for chL(λ) in this case Finally when g is non-symmetrizable itis unknown how to compute weight space multiplicities even for integrable L(λ) Thus for gnon-symmetrizable our formulae provide as much information on chL(λ) as one could hopefor given existing methods

We obtain the above three formulae using the following theorem For V a general highestweight module it gives a necessary and sufficient condition for M(λ IV ) rarr V to induce anequality of weights in terms of the action of a certain Levi subalgebra Define the potentialintegrability of V to be IpV = IL(λ) IV To justify the terminology note that these areprecisely the simple directions which become integrable in quotients of V

Theorem 28 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Theorem 28 is new in both finite and infinite type

23 A question of Lepowsky on the weights of highest weight modules In thissection we revisit the problem encountered in the proofs of Propositions 22 24 26 namelyunder what circumstances does the datum of Theorem 12 determine wtV

It was noticed by the second named author in [28] that wtV is a finer invariant thanconv V whenever the potential integrability IpV contains orthogonal roots The prototypicalexample here is for g = sl2 oplus sl2 and V = M(0)M(s1s2 middot 0)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 5

James Lepowsky asked whether this orthogonality is the only obstruction to determiningwtV from its convex hull In finite type this would reduce to a question in types A2 B2and G2 We now answer this question affirmatively for g an arbitrary KacndashMoody algebra

Theorem 29 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if IL(λ) J is complete ie (αi αj) 6=0 foralli j isin IL(λ) J

We are not aware of a precursor to Theorem 29 in the literature even in finite type

24 The WeylndashKac formula for the weights of simple modules We prove the fol-lowing identity whose notation we will explain below

Theorem 210 For λ isin hlowast such that the stabilizer of λ in WIL(λ)is finite we have

wtL(λ) =sum

wisinWIL(λ)

weλprod

iisinI(1minus eminusαi) (211)

On the left hand side of (211) we mean the lsquomultiplicity-freersquo charactersum

microisinhlowastL(λ)micro 6=0 emicro

On the right hand side of (211) in each summand we take the lsquohighest weightrsquo expansion ofw(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0

In particular as claimed in the introduction each summand comes with signs but no multi-plicities as in KazhdanndashLusztig theory

Precursors to Theorem 210 include Kass [26] Brion [8] Walton [42] Postnikov [39] andSchutzer [40] for the integrable highest weight modules As pointed out to us by MichelBrion in finite type Theorem 210 follows from a more general formula for exponential sumsover polyhedra cf Remark 79 below All other cases are to our knowledge new

25 A question of Brion on localization of D-modules and a geometric interpre-tation of integrability In this section we use a topological manifestation of the datum inTheorem 12 to answer a question of Brion in geometric representation theory

Brionrsquos question is as follows Let g be of finite type λ a regular dominant integral weightand consider convL(λ) the Weyl polytope For any J sub I if one intersects the tangentcones of convL(λ) at wλw isin WJ one obtains convM(λ J) Brion first asked whetheran analogous formula holds for other highest weight modules We answer this questionaffirmatively in [11]

Brion further observed that this procedure of localizing convL(λ) in convex geometry isthe shadow of a localization in complex geometry More precisely let X denote the flagvariety and Lλ denote the line bundle on X with H0(XLλ) L(λ) As usual write Xw

for the closure of the Bruhat cell Cw = BwBB forallw isin W and write w for the longestelement of W Let D denote the standard dualities on Category O and regular holonomicD-modules Then for a union of Schubert divisors Z = cupiisinIJXsiw with complement U

we have H0(ULλ) DM(λ J) Thus in this case localization of the convex hull could berecovered as taking the convex hull of the weights of sections of Lλ on an appropriate openset U Brion asked whether a similar result holds for more general highest weight modules

We answer this affirmatively for a regular integral infinitesimal character By translationit suffices to examine the regular block O0

6 GURBIR DHILLON AND APOORVA KHARE

Theorem 212 Let g be of finite type and λ = wmiddotminus2ρ Let V be a g-module of highest weightλ and write V for the corresponding D-module on X For J sub IV set Z = cupiisinIV JXsiw

and write GB = Z t U Then we have DH0(U DV) is a g-module of highest weight λ andintegrability J

Let us mention one ingredient in the proof of Theorem 212 Let Vrh denote the perversesheaf corresponding to V under the RiemannndashHilbert correspondence Then the datum ofTheorem 12 manifests as

Proposition 213 For J sub IL(λ) consider the smooth open subvariety of Xw given oncomplex points by

UJ = Cw t⊔iisinJ

Csiw (214)

Letting VVrh be as above upon restricting to UIL(λ)we have

Vrh|UIL(λ) jCUIV [`(w)] (215)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

To our knowledge Proposition 213 is new though not difficult to prove once stated Theproof is a pleasing geometric avatar of the idea that M(λ IV ) rarr V is an isomorphism lsquotofirst orderrsquo

26 Highest weight modules over symmetrizable quantum groups In the final sec-tion we apply both the methods and the results from earlier to study highest weight modulesover quantum groups and obtain results similar to those discussed above

27 Further problems We conclude this section by calling attention to two problemssuggested by our results

Problem 1 WeylndashKac formula with infinite stabilizers The statement of Theorem 210includes the assumption that the highest weight has finite integrable stabilizer However theasserted identity otherwise tends to fail in interesting ways For example

Proposition 216 For g of rank 2 and the trivial module L(0) we havesumwisinW

w1

(1minus eminusα1)(1minus eminusα2)= 1 +

sumαisin∆im

minus

eα (217)

where ∆imminus denotes the set of negative imaginary roots

It would be interesting to formulate a correct version of Theorem 210 even for one-dimensionalmodules in affine type which might be thought of as a multiplicity-free Macdonald identityIndeed Equation (217) suggests correction terms coming from imaginary roots akin to [24]

Problem 2 From convex hulls to weights The regular block of Category O is the prototypi-cal example of a category of infinite-dimensional g-modules of perverse sheaves on a stratifiedspace and of a highest weight category Accordingly it has very rich representation-theoreticgeometric and purely categorical structure The main theorem of the paper Theorem 12determines the meaning of the convex hull of the weights in these three guises namely theintegrability behavior in codimension one and the JordanndashHolder content [V L(si middot λ)] Itwould be very interesting to obtain a similar understanding of the weights themselves

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7

Acknowledgments It is a pleasure to thank Michel Brion Daniel Bump Shrawan KumarJames Lepowsky Sam Raskin Geordie Williamson and Zhiwei Yun for valuable discussionsThe work of GD was partially supported by the Department of Defense (DoD) through theNDSEG fellowship

3 Preliminaries and notations

The contents of this section are mostly standard We advise the reader to skim Subsection33 and refer back to the rest only as needed

31 Notation for numbers and sums We write Z for the integers and QRC for therational real and complex numbers respectively For a subset S of a real vector space E wewrite Zgt0S for the set of finite linear combinations of S with coefficients in Zgt0 and similarlyZSQgt0SRS etc

32 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [25] and [33] In this paper we work throughout over C Let I bea finite set and A = (aij)ijisinI a generalized Cartan matrix Fix a realization (h π π) withsimple roots π = αiiisinI sub hlowast and coroots π = αiiisinI sub h satisfying (αi αj) = aij foralli j isin I

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h αi)ei [h fi] = minus(h αi)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially thesecoincide when A is symmetrizable When A is clear from context we will abbreviate theseto g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 forα isin ∆+ and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0

write suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin Jand h For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λ Writing AJfor the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJ as a

subalgebra of g(A) Now write πJ ∆+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finallywe define the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 5

James Lepowsky asked whether this orthogonality is the only obstruction to determiningwtV from its convex hull In finite type this would reduce to a question in types A2 B2and G2 We now answer this question affirmatively for g an arbitrary KacndashMoody algebra

Theorem 29 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if IL(λ) J is complete ie (αi αj) 6=0 foralli j isin IL(λ) J

We are not aware of a precursor to Theorem 29 in the literature even in finite type

24 The WeylndashKac formula for the weights of simple modules We prove the fol-lowing identity whose notation we will explain below

Theorem 210 For λ isin hlowast such that the stabilizer of λ in WIL(λ)is finite we have

wtL(λ) =sum

wisinWIL(λ)

weλprod

iisinI(1minus eminusαi) (211)

On the left hand side of (211) we mean the lsquomultiplicity-freersquo charactersum

microisinhlowastL(λ)micro 6=0 emicro

On the right hand side of (211) in each summand we take the lsquohighest weightrsquo expansion ofw(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0

In particular as claimed in the introduction each summand comes with signs but no multi-plicities as in KazhdanndashLusztig theory

Precursors to Theorem 210 include Kass [26] Brion [8] Walton [42] Postnikov [39] andSchutzer [40] for the integrable highest weight modules As pointed out to us by MichelBrion in finite type Theorem 210 follows from a more general formula for exponential sumsover polyhedra cf Remark 79 below All other cases are to our knowledge new

25 A question of Brion on localization of D-modules and a geometric interpre-tation of integrability In this section we use a topological manifestation of the datum inTheorem 12 to answer a question of Brion in geometric representation theory

Brionrsquos question is as follows Let g be of finite type λ a regular dominant integral weightand consider convL(λ) the Weyl polytope For any J sub I if one intersects the tangentcones of convL(λ) at wλw isin WJ one obtains convM(λ J) Brion first asked whetheran analogous formula holds for other highest weight modules We answer this questionaffirmatively in [11]

Brion further observed that this procedure of localizing convL(λ) in convex geometry isthe shadow of a localization in complex geometry More precisely let X denote the flagvariety and Lλ denote the line bundle on X with H0(XLλ) L(λ) As usual write Xw

for the closure of the Bruhat cell Cw = BwBB forallw isin W and write w for the longestelement of W Let D denote the standard dualities on Category O and regular holonomicD-modules Then for a union of Schubert divisors Z = cupiisinIJXsiw with complement U

we have H0(ULλ) DM(λ J) Thus in this case localization of the convex hull could berecovered as taking the convex hull of the weights of sections of Lλ on an appropriate openset U Brion asked whether a similar result holds for more general highest weight modules

We answer this affirmatively for a regular integral infinitesimal character By translationit suffices to examine the regular block O0

6 GURBIR DHILLON AND APOORVA KHARE

Theorem 212 Let g be of finite type and λ = wmiddotminus2ρ Let V be a g-module of highest weightλ and write V for the corresponding D-module on X For J sub IV set Z = cupiisinIV JXsiw

and write GB = Z t U Then we have DH0(U DV) is a g-module of highest weight λ andintegrability J

Let us mention one ingredient in the proof of Theorem 212 Let Vrh denote the perversesheaf corresponding to V under the RiemannndashHilbert correspondence Then the datum ofTheorem 12 manifests as

Proposition 213 For J sub IL(λ) consider the smooth open subvariety of Xw given oncomplex points by

UJ = Cw t⊔iisinJ

Csiw (214)

Letting VVrh be as above upon restricting to UIL(λ)we have

Vrh|UIL(λ) jCUIV [`(w)] (215)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

To our knowledge Proposition 213 is new though not difficult to prove once stated Theproof is a pleasing geometric avatar of the idea that M(λ IV ) rarr V is an isomorphism lsquotofirst orderrsquo

26 Highest weight modules over symmetrizable quantum groups In the final sec-tion we apply both the methods and the results from earlier to study highest weight modulesover quantum groups and obtain results similar to those discussed above

27 Further problems We conclude this section by calling attention to two problemssuggested by our results

Problem 1 WeylndashKac formula with infinite stabilizers The statement of Theorem 210includes the assumption that the highest weight has finite integrable stabilizer However theasserted identity otherwise tends to fail in interesting ways For example

Proposition 216 For g of rank 2 and the trivial module L(0) we havesumwisinW

w1

(1minus eminusα1)(1minus eminusα2)= 1 +

sumαisin∆im

minus

eα (217)

where ∆imminus denotes the set of negative imaginary roots

It would be interesting to formulate a correct version of Theorem 210 even for one-dimensionalmodules in affine type which might be thought of as a multiplicity-free Macdonald identityIndeed Equation (217) suggests correction terms coming from imaginary roots akin to [24]

Problem 2 From convex hulls to weights The regular block of Category O is the prototypi-cal example of a category of infinite-dimensional g-modules of perverse sheaves on a stratifiedspace and of a highest weight category Accordingly it has very rich representation-theoreticgeometric and purely categorical structure The main theorem of the paper Theorem 12determines the meaning of the convex hull of the weights in these three guises namely theintegrability behavior in codimension one and the JordanndashHolder content [V L(si middot λ)] Itwould be very interesting to obtain a similar understanding of the weights themselves

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7

Acknowledgments It is a pleasure to thank Michel Brion Daniel Bump Shrawan KumarJames Lepowsky Sam Raskin Geordie Williamson and Zhiwei Yun for valuable discussionsThe work of GD was partially supported by the Department of Defense (DoD) through theNDSEG fellowship

3 Preliminaries and notations

The contents of this section are mostly standard We advise the reader to skim Subsection33 and refer back to the rest only as needed

31 Notation for numbers and sums We write Z for the integers and QRC for therational real and complex numbers respectively For a subset S of a real vector space E wewrite Zgt0S for the set of finite linear combinations of S with coefficients in Zgt0 and similarlyZSQgt0SRS etc

32 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [25] and [33] In this paper we work throughout over C Let I bea finite set and A = (aij)ijisinI a generalized Cartan matrix Fix a realization (h π π) withsimple roots π = αiiisinI sub hlowast and coroots π = αiiisinI sub h satisfying (αi αj) = aij foralli j isin I

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h αi)ei [h fi] = minus(h αi)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially thesecoincide when A is symmetrizable When A is clear from context we will abbreviate theseto g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 forα isin ∆+ and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0

write suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin Jand h For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λ Writing AJfor the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJ as a

subalgebra of g(A) Now write πJ ∆+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finallywe define the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

6 GURBIR DHILLON AND APOORVA KHARE

Theorem 212 Let g be of finite type and λ = wmiddotminus2ρ Let V be a g-module of highest weightλ and write V for the corresponding D-module on X For J sub IV set Z = cupiisinIV JXsiw

and write GB = Z t U Then we have DH0(U DV) is a g-module of highest weight λ andintegrability J

Let us mention one ingredient in the proof of Theorem 212 Let Vrh denote the perversesheaf corresponding to V under the RiemannndashHilbert correspondence Then the datum ofTheorem 12 manifests as

Proposition 213 For J sub IL(λ) consider the smooth open subvariety of Xw given oncomplex points by

UJ = Cw t⊔iisinJ

Csiw (214)

Letting VVrh be as above upon restricting to UIL(λ)we have

Vrh|UIL(λ) jCUIV [`(w)] (215)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

To our knowledge Proposition 213 is new though not difficult to prove once stated Theproof is a pleasing geometric avatar of the idea that M(λ IV ) rarr V is an isomorphism lsquotofirst orderrsquo

26 Highest weight modules over symmetrizable quantum groups In the final sec-tion we apply both the methods and the results from earlier to study highest weight modulesover quantum groups and obtain results similar to those discussed above

27 Further problems We conclude this section by calling attention to two problemssuggested by our results

Problem 1 WeylndashKac formula with infinite stabilizers The statement of Theorem 210includes the assumption that the highest weight has finite integrable stabilizer However theasserted identity otherwise tends to fail in interesting ways For example

Proposition 216 For g of rank 2 and the trivial module L(0) we havesumwisinW

w1

(1minus eminusα1)(1minus eminusα2)= 1 +

sumαisin∆im

minus

eα (217)

where ∆imminus denotes the set of negative imaginary roots

It would be interesting to formulate a correct version of Theorem 210 even for one-dimensionalmodules in affine type which might be thought of as a multiplicity-free Macdonald identityIndeed Equation (217) suggests correction terms coming from imaginary roots akin to [24]

Problem 2 From convex hulls to weights The regular block of Category O is the prototypi-cal example of a category of infinite-dimensional g-modules of perverse sheaves on a stratifiedspace and of a highest weight category Accordingly it has very rich representation-theoreticgeometric and purely categorical structure The main theorem of the paper Theorem 12determines the meaning of the convex hull of the weights in these three guises namely theintegrability behavior in codimension one and the JordanndashHolder content [V L(si middot λ)] Itwould be very interesting to obtain a similar understanding of the weights themselves

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7

Acknowledgments It is a pleasure to thank Michel Brion Daniel Bump Shrawan KumarJames Lepowsky Sam Raskin Geordie Williamson and Zhiwei Yun for valuable discussionsThe work of GD was partially supported by the Department of Defense (DoD) through theNDSEG fellowship

3 Preliminaries and notations

The contents of this section are mostly standard We advise the reader to skim Subsection33 and refer back to the rest only as needed

31 Notation for numbers and sums We write Z for the integers and QRC for therational real and complex numbers respectively For a subset S of a real vector space E wewrite Zgt0S for the set of finite linear combinations of S with coefficients in Zgt0 and similarlyZSQgt0SRS etc

32 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [25] and [33] In this paper we work throughout over C Let I bea finite set and A = (aij)ijisinI a generalized Cartan matrix Fix a realization (h π π) withsimple roots π = αiiisinI sub hlowast and coroots π = αiiisinI sub h satisfying (αi αj) = aij foralli j isin I

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h αi)ei [h fi] = minus(h αi)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially thesecoincide when A is symmetrizable When A is clear from context we will abbreviate theseto g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 forα isin ∆+ and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0

write suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin Jand h For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λ Writing AJfor the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJ as a

subalgebra of g(A) Now write πJ ∆+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finallywe define the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7

Acknowledgments It is a pleasure to thank Michel Brion Daniel Bump Shrawan KumarJames Lepowsky Sam Raskin Geordie Williamson and Zhiwei Yun for valuable discussionsThe work of GD was partially supported by the Department of Defense (DoD) through theNDSEG fellowship

3 Preliminaries and notations

The contents of this section are mostly standard We advise the reader to skim Subsection33 and refer back to the rest only as needed

31 Notation for numbers and sums We write Z for the integers and QRC for therational real and complex numbers respectively For a subset S of a real vector space E wewrite Zgt0S for the set of finite linear combinations of S with coefficients in Zgt0 and similarlyZSQgt0SRS etc

32 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [25] and [33] In this paper we work throughout over C Let I bea finite set and A = (aij)ijisinI a generalized Cartan matrix Fix a realization (h π π) withsimple roots π = αiiisinI sub hlowast and coroots π = αiiisinI sub h satisfying (αi αj) = aij foralli j isin I

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h αi)ei [h fi] = minus(h αi)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially thesecoincide when A is symmetrizable When A is clear from context we will abbreviate theseto g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 forα isin ∆+ and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0

write suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin Jand h For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λ Writing AJfor the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJ as a

subalgebra of g(A) Now write πJ ∆+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finallywe define the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

8 GURBIR DHILLON AND APOORVA KHARE

33 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and theTits cone as C =

⋃wisinW wD

Remark 31 In [33] and [25] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [33 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0 forallj isin J The following standard properties will be usedwithout further reference in the paper

Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with real-ization (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isin WJ wmicro = micro is generated by the simplereflections it contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the JTits cone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ

is a convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is

an interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

Proof The reader can easily check that the standard arguments cf [33 Proposition 142]apply in this setting mutatis mutandis

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W viaw middot micro = w(micro+ ρ)minus ρ this does not depend on the choice of ρ

34 Representations integrability and parabolic Verma modules Given an h-module M and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro =m isinM hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Let V be a highest weight g-module with highest weight λ isin hlowast For J sub I we say V is Jintegrable if fj acts locally nilpotently on V forallj isin J The following standard lemma may bededuced from [33 Lemma 133] and the proof of [33 Lemma 135]

Lemma 33

(1) If V is of highest weight λ then V is J integrable if and only if fj acts nilpotently onthe highest weight line Vλ forallj isin J

(2) If V is J integrable then chV is WJ -invariant

Let IV denote the maximal J for which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the originalpapers by Lepowsky (see [35] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module From the definition andLemma 33 it follows that M(λ J) is a highest weight module that is J integrable

More generally for any subalgebra lJ sub s sub g the module Ms(λ J) will co-represent thefunctor from s -mod to Set

M m isinMλ (n+ cap s)m = 0 fj acts nilpotently on mforallj isin J (36)

We will only be concerned with s equal to a Levi or parabolic subalgebra In accordance withthe literature in the case of ldquofull integrabilityrdquo we define Lmax(λ) = M(λ I) for λ isin P+and similarly Lmax

pJ(λ) = MpJ (λ J) and Lmax

lJ(λ) = MlJ (λ J) for λ isin P+

J Finally write

LlJ (λ) for the simple quotient of LmaxlJ

(λ)

Proposition 37 (Basic formulae for parabolic Verma modules) Let λ isin P+J

(1) M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J)

(2) Given a Lie subalgebra lJ sub s sub g define s+J = lJ oplus (u+

J cap s) Then

Ms(λ J) Indss+J

ReslJs+JLmaxlJ

(λ) (38)

where we view lJ as a quotient of s+J via the short exact sequence

0rarr u+J cap srarr s+

J rarr lJ rarr 0

In particular for s = g

wtM(λ J) = wtLlJ (λ)minus Zgt0(∆+ ∆+J ) (39)

We remark that some authors use the term inflation (from lJ to s+J ) in place of ReslJ

s+J

4 A classification of highest weight modules to first order

In this section we prove the main result of the paper Theorem 12 except for the perversesheaf formulation (4) which we address in Section 8 The two tools we use are (i) anldquoIntegrable Slice Decompositionrdquo of the weights of parabolic Verma modules M(λ J) whichextends a previous construction in [28] to representations of KacndashMoody algebras and (ii) aldquoRay Decompositionrdquo of the convex hull of these weights which is novel in both finite andinfinite type for non-integrable highest weight modules

41 The Integrable Slice Decomposition The following is the technical heart of thissection

Proposition 41 (Integrable Slice Decomposition)

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (42)

In particular wtM(λ J) lies in the J Tits cone (cf Section 33)

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

10 GURBIR DHILLON AND APOORVA KHARE

In proving Proposition 41 and below the following results will be of use to us

Proposition 43 ([25 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite

type then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ(3) wtV = (λminus Zgt0π) cap conv(Wλ)

Proposition 43 is explicitly stated in [25] for g but also holds for g To see this note thatwtV sub (λ minus Zgt0π) cap conv(Wλ) and the latter are the weights of its simple quotient L(λ)which is inflated from g

Proof of Proposition 41 The disjointness of the terms on the right-hand side is an easyconsequence of the linear independence of simple roots

We first show the inclusion sup Recall the isomorphism of Proposition 37

M(λ J) M(λ)(f(αj λ)+1j M(λ)λ forallj isin J)

It follows that the weights of ker(M(λ) rarr M(λ J)) are contained in⋃jisinJν 6 sj middot λ

Hence λ minus micro is a weight of M(λ J)forallmicro isin Zgt0πIJ Any nonzero element of M(λ J)λminusmicrogenerates an integrable highest weight lJ -module As the weights of all such modules coincideby Proposition 43(3) this shows the inclusion sup

We next show the inclusion sub For any micro isin Zgt0πIJ the lsquointegrable slicersquo

Smicro =oplus

νisinλminusmicro+ZπJ

M(λ J)ν

lies in Category O for lJ and is furthermore an integrable lJ -module It follows that theweights of M(λ J) lie in the J Tits cone

Let ν be a weight of M(λ J) and write λminus ν = microJ + microIJ microJ isin Zgt0πJ microIJ isin Zgt0πIJ We need to show ν isin wtLlJ (λminus microIJ) By WJ -invariance we may assume ν is J dominantBy Proposition 43 it suffices to show that ν is non-degenerate with respect to (λ minus microIJ)To see this using Proposition 37 write

ν = λminus microL minusnsumk=1

βk where λminus microL isin wtLlJ (λ) βk isin ∆+ ∆+J 1 6 k 6 n

The claimed nondegeneracy follows from the fact that λminus microL is nondegenerate with respectto λ and that the support of each βk is connected

As an immediate consequence of the Integrable Slice Decomposition 41 we present a familyof decompositions of wtM(λ J) which interpolates between the two sides of Equation (42)

Corollary 44 For subsets J sub J prime sub I we have

wtM(λ J) =⊔

microisinZgt0(ππJprime )

wtMlJprime (λminus micro J) (45)

where MlJprime (λ J) was defined in Equation (36)

As a second consequence Proposition 43(2) and the Integrable Slice Decomposition 41yield the following simple description of the weights of most parabolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 46 Suppose λ has finite WJ -isotropy Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6 λ (47)

Equipped with the Integrable Slice Decomposition we provide a characterization of theweights of a parabolic Verma module that will be helpful in Section 5

Proposition 48wtM(λ J) = (λ+ Zπ) cap convM(λ J) (49)

Proof The inclusion sub is immediate For the reverse sup note that convM(λ J) lies in theJ Tits cone and is WJ invariant It then suffices to consider a point ν of the right-hand sidewhich is J dominant Write ν as a convex combination

ν =nsumk=1

tk(λminus microkIJ minus microkJ)

where tk isin Rgt0sum

k tk = 1 microkIJ isin Zgt0πIJ microkJ isin Zgt0πJ 1 6 k 6 n By Propositions 43

and 41 it remains to observe that ν is non-degenerate with respect to λ minussum

k tkmicrokIJ as a

similar statement holds for each summand

42 The Ray Decomposition Using the Integrable Slice Decomposition we obtain thefollowing novel description of convM(λ J) which is of use in the proof of the main theoremand throughout the paper

Proposition 410 (Ray Decomposition)

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) (411)

When J = I by the right-hand side we mean conv⋃wisinW wλ

Proof The containment sup is straightforward using the definition of integrability and WJ

invariance For the containment sub it suffices to show every weight of M(λ J) lies in theright-hand side Note that the right-hand side contains WJ(λminusZgt0πIJ) so we are done bythe Integrable Slice Decomposition 41

43 Proof of the main result With the Integrable Slice and Ray Decompositions in handwe now turn to our main theorem

Theorem 412 Given a highest weight module V and a subset J sub I the following areequivalent

(1) J = IV (2) conv wtV = conv wtM(λ J)(3) The stabilizer of conv V in W is WJ

The connection to perverse sheaves promised in Section 1 is proven in Proposition 82

Proof To show (1) implies (2) note that λ minus Zgt0αi sub wtVforalli isin πIIV The implicationnow follows from the Ray Decomposition 410 The remaining implications follow from theassertion that the stabilizer of conv V in W is WIV Thus it remains to prove the assertionIt is standard that WIV preserves conv V For the reverse since (1) implies (2) we mayreduce to the case of V = M(λ J) Suppose w isinW stabilizes convM(λ J) It is easy to seeλ is a face of convM(λ J) hence so is wλ However by the Ray Decomposition 410 it is

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

12 GURBIR DHILLON AND APOORVA KHARE

clear that the only 0-faces of convM(λ J) are WJ(λ) so without loss of generality we mayassume w stabilizes λ

Recalling that WJ is exactly the subgroup of W which preserves ∆+∆+J it suffices to show

w preserves ∆+ ∆+J Let α isin ∆+ ∆+

J then by Proposition 37 λ minus Zgt0α sub wtM(λ J)whence so is λ minus Zgt0w(α) by Proposition 48 This shows that w(α) gt 0 it remains toshow w(α) isin ∆+

J It suffices to check this for the simple roots αi i isin I J Suppose not

ie w(αi) isin ∆+J for some i isin I J In this case note w(αi) must be a real root of ∆+

J egby considering 2w(αi) Thus the w(αi) root string λ minus Zgt0w(αi) is the set of weights of anintegrable representation of gminusw(αi) oplus [gminusw(αi) gw(αi)]oplus gw(αi) sl2 which is absurd

Corollary 413 The stabilizer of chV in W is WIV

Corollary 414 For any V conv V is the Minkowski sum of convWIV (λ) and the coneRgt0WIV (πIIV )

Proof By Theorem 412 we reduce to the case of V a parabolic Verma module Now theresult follows from the Ray Decomposition 410 and Equation (39)

Remark 415 For g semisimple the main theorem 412 and the Ray Decomposition 410imply that the convex hull of weights of any highest weight module V is a WIV -invariantpolyhedron To our knowledge this was known for many but not all highest weight modulesby recent work [28] and prior to that only for parabolic Verma modules We develop manyother applications of the main theorem to the convex geometry of conv V including theclassification of its faces and their inclusions in the companion work [11]

5 A question of Bump and weight formulas for non-integrable simplemodules

We remind the three positive formulas for the weights of simple highest weight modules tobe obtained in this section

Proposition 51 Write l for the Levi subalgebra corresponding to IL(λ) and write Ll(ν) forthe simple l module of highest weight ν isin hlowast Then

wtL(λ) =⊔

microisinZgt0ππIL(λ)

wtLl(λminus micro) (52)

Proposition 53

wtL(λ) = convL(λ) cap micro isin hlowast micro 6 λ (54)

Proposition 55 Suppose λ has finite stabilizer in WIL(λ) Then

wtL(λ) = WIL(λ)micro isin P+

IL(λ) micro 6 λ (56)

Remark 57 For an extension of Proposition 55 to arbitrary λ see the companion work[11]

Note that the three weight formulas above are immediate consequences of combining thefollowing theorem with Propositions 41 and 48 and Corollary 46 respectively Recall fromSection 22 that for a highest weight module V we defined its potential integrability to beIpV = IL(λ) IV

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Theorem 58 Let V be a highest weight module Vλ its highest weight line Let l denote theLevi subalgebra corresponding to IpV Then wtV = wtM(λ IV ) if and only if wtU(l)Vλ =wtU(l)M(λ IV )λ ie wtU(l)Vλ = λminus Zgt0πIpV

In particular if V = L(λ) then wtL(λ) = wtM(λ IL(λ))

Proof It is clear eg by the Integrable Slice Decomposition 41 that if wtV = wtM(λ IV )then wtIpV

V = λ minus Zgt0πIpV For the reverse implication by again using the Integrable Slice

Decomposition and the action of lIV it suffices to show that λminus Zgt0πIIV sub wtV

Suppose not ie there exists micro isin Zgt0πIIV with Vλminusmicro = 0 We decompose I IV = IpV tIqand write accordingly micro = microp + microq where microp isin Zgt0πIpV

and microq isin Zgt0πIq By assumption

Vλminusmicrop is nonzero so by the PBW theorem there exists part isin U(nminusIpV

) of weight minusmicrop such that

part middotVλ is nonzero Now choose an enumeration of Iq = ik 1 6 k 6 n write microq =sum

kmkαik and consider the monomials E =

prodk e

mkik F =

prodk f

mkik

By the vanishing of Vλminusmicropminusmicroq partF

annihilates the highest weight line Vλ whence so does EpartF Since [ei fj ] = 0 foralli 6= j wemay write this as part

prodk e

mkikfmkik

and each factor emkik fmkik

acts on Vλ by a nonzero scalar acontradiction

Remark 59 Given the delicacy of Jantzenrsquos criterion for the simplicity of a parabolic Vermamodule (see [21 22]) it is interesting that the equality on weights of L(λ) and M(λ IL(λ))always holds

6 A question of Lepowsky on the weights of highest weight modules

In Section 5 we applied our main result to obtain formulas for the weights of simplemodules We now explore the extent to which this can be done for arbitrary highest weightmodules thereby answering the question of Lepowsky discussed in Section 23

Theorem 61 Fix λ isin hlowast and J sub IL(λ) Every highest weight module V of highest weightλ and integrability J has the same weights if and only if Jp = IL(λ) J is complete ie(αj αjprime) 6= 0forallj jprime isin Jp

We first sketch our approach The subcategory of modules V with fixed highest weightλ and integrability J is basically a poset modulo scaling The source here is the parabolicVerma module M(λ J) and we are concerned with the possible vanishing of weight spaces aswe move away from the source The most interesting ingredient of the proof is the observationthat there is a sink which we call L(λ J) and hence the question reduces to whether M(λ J)and L(λ J) have the same weights

Lemma 62 For J sub IL(λ) there is a minimal quotient L(λ J) of M(λ) satisfying theequivalent conditions of Theorem 412

Proof Consider all submodules N of M(λ) such that NλminusZgt0αi = 0 foralli isin I J There is amaximal such namely their sum N prime and it is clear that L(λ J) = M(λ)N prime by construction

As the reader is no doubt aware the inexplicit construction of L(λ J) here is parallel tothat of L(λ) ndash in fact L(λ) = L(λ IL(λ)) The existence of the objects L(λ J) was noted bythe second named author in [28] but we are unaware of other appearances in the literatureIn [28] the character of L(λ J) was not determined However it turns out to be no moredifficult than chL(λ) under the following sufficient condition which as we explain below isoften satisfied

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

14 GURBIR DHILLON AND APOORVA KHARE

Proposition 63 Fix λ isin hlowast J sub IL(λ) Suppose that Ext1O(L(λ) L(si middot λ)) 6= 0 foralli isin

IL(λ) J Then there is a short exact sequence

0rarroplus

iisinIL(λ)J

L(si middot λ)rarr L(λ J)rarr L(λ)rarr 0 (64)

Proof The choice of a nonzero class in each Ext1O(L(λ) L(si middot λ)) i isin IL(λ) J gives an ex-

tension E as in Equation (64) The space Eλ is a highest weight line and we obtain anassociated map M(λ) M(λ) otimes Eλ rarr E This is surjective as follows from an easy argu-ment using JordanndashHolder content and the nontriviality of each extension The consequentsurjection E rarr L(λ J) is an isomorphism by considering JordanndashHolder content

Remark 65 By identifying Ext1O(L(λ) L(si middot λ)) with Hom(N(λ) L(si middot λ)) where N(λ)

is the maximal submodule of M(λ) one sees it is at most one dimensional It is nonzero inthe following two situations First when g is symmetrizable and λ is dominant integral hereone uses the end of the BGG resolution Second when g is of finite type and λ is arbitrarywith si middotλ 6 λ In this case by work of Soergel [41] (see also [20 sect34]) it suffices to considerλ integral in which case this is a standard consequence of KazhdanndashLusztig theory See [20Theorem 815(c)] for λ regular and Irving [19 Corollary 135] for λ singular We sketch aproof in the regular case by considering the corresponding ICs on GB By parity vanishingthe Cousin spectral sequence for RHom associated to the Schubert stratification collapseson E1 cf [5 sect34] This reduces the question to only the Schubert strata indexed by si middot λ λwhere it is clear

We expect the hypotheses in Proposition 63 to hold in most cases of interest eg for λ inthe dot orbit of a dominant integral weight

Returning to Lepowskyrsquos question for g possibly non-symmetrizable we first prove thefollowing weaker form of Equation (64)

Proposition 66 Let g be of arbitrary type and λ isin P+ Then

wtL(λ J) = wtL(λ) cup⋃iisinIJ

wtL(si middot λ) (67)

Proof The restriction of chM(λ) to the root string λminusZgt0αi i isin I J is entirely determinedby the JordanndashHolder content [M(λ) L(λ)] = [M(λ) L(si middot λ)] = 1 Using this it sufficesto exhibit a module with weights given by the right-hand side of (67)

Consider the end of the BGG resolutionoplusiisinI

M(si middot λ)rarrM(λ)rarr Lmax(λ)rarr 0 (68)

For each i isin I let N(si middotλ) denote the maximal submodule of M(si middotλ) Consider the quotientQ of M(λ) by the image of

oplusjisinJM(sj middot λ)oplus

oplusiisinIJ N(si middot λ) under (68) We deduce

chQ = chLmax(λ) +sumiisinIJ

chL(si middot λ)

which implies that wtQ whence wtL(λ J) coincides with the right-hand side of (67)

We now answer Lepowskyrsquos question By the above results it suffices to determine whenwtL(λ J) = wtM(λ J)

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

Proof of Theorem 61 It will be clarifying though not strictly necessary for the argumentto observe the following compatibility of the construction of L(λ J) and restriction to a Levi

Lemma 69 Let I prime sub I l = lIprime the corresponding Levi subalgebra and for J sub I writeJ prime = J cap I prime Then

Ll(λ Jprime) U(l)L(λ J)λ

where Ll(λ Jprime) is the smallest l-module with integrability J prime constructed as in Lemma 62

Proof Since the integrability of U(l)L(λ J)λ is J prime we have a surjection

U(l)L(λ J)λ rarr Ll(λ Jprime)rarr 0 (610)

To see that Equation (610) is an isomorphism we need to show the kernel K of Ml(λ Jprime)rarr

Ll(λ Jprime) is sent into the kernel of M(λ J) rarr L(λ J) under the inclusion Ml(λ J

prime) rarrM(λ J) To do so choose a filtration of K

0 sube K1 sube K2 sube middot middot middot K = cupmgt0Km

where KmKmminus1 is generated as a U(l)-module by a highest weight vector vmIt suffices to prove by induction on m that the U(g)-module generated by Km lies in the ker-

nel of M(λ J)rarr L(λ J) forallm gt 0 For the inductive step suppose that M(λ J)U(g)Kmminus1

has integrability J Then choosing a lift vm of vm to Km observe that vm is a highest weightvector in M(λ J)U(g)Kmminus1 for the action of all of U(g) by the linear independence of sim-ple roots It is clear that U(g)vm has trivial intersection with the (λminusZgt0αi)-weight spaces ofM(λ J)U(g)Kmminus1 for all i isin I J hence M(λ J)U(g)Km again has integrability J

We are now ready to prove Theorem 61 By Theorem 58 we have wtL(λ J) = wtM(λ J)if and only if wtJp L(λ J) = wtJpM(λ J) By Lemma 69 we may therefore assume thatJ = empty and λ isin P+ and hence Jp = I

By Proposition 66 we have

wtL(λ empty) = wtL(λ) cup⋃iisinI

wtL(si middot λ)

wtL(λ I i) = wtL(λ) cup wtL(si middot λ) foralli isin I

Combining the two above equalities we have

wtL(λ empty) =⋃iisinI

wtL(λ I i)

For L(λ I i) the set of potentially integrable directions is i so we may apply Theorem58 to conclude

wtL(λ empty) =⋃iisinI

wtM(λ I i) (611)

Thus it remains to show that for λ dominant integral wtM(λ) =⋃iisinI wtM(λ I i) if and

only if the Dynkin diagram of g is completeSuppose first that the Dynkin diagram of g is not complete Pick i iprime isin I with (αi αiprime) = 0

Then M(λ) and M(λ)sisiprime middotM(λ)λ have distinct weights as can be seen by a calculation intype A1 timesA1

Suppose now that the Dynkin diagram of g is complete By (611) it suffices to show

wtM(λ I i) sup λminussumjisinI

kjαj kj isin Zgt0 ki gt kj forallj isin I

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

16 GURBIR DHILLON AND APOORVA KHARE

But this follows using the assumption that every node is connected to i and the representationtheory of sl2

Remark 612 Theorem 61 can be proved without working directly with the objects L(λ J)but instead using only the lower bound (67) for the weights

wtV sup wtL(λ IV ) = wtL(λ) cup⋃

iisinIL(λ)IV

wtL(si middot λ)

7 The WeylndashKac formula for the weights of simple modules

The main goal of this section is to prove the following

Theorem 71 Let λ isin hlowast be such that the stabilizer of λ in WIL(λ)is finite Then

wtL(λ) =sum

wisinWIL(λ)

weλprod

αisinπ(1minus eminusα) (72)

We remind that on the left hand side of (72) we mean the lsquomultiplicity-freersquo charactersummicroisinhlowastL(λ)micro 6=0 e

micro On the right hand side of (72) in each summand we take the lsquohighest

weightrsquo expansion of w(1minus eminusαi)minus1 ie

w1

1minus eminusαi=

1 + eminuswαi + eminus2wαi + middot middot middot wαi gt 0

minusewαi minus e2wαi minus e3wαi minus middot middot middot wαi lt 0(73)

By Theorem 58 we will deduce Theorem 71 from the following

Proposition 74 Fix λ isin hlowast J sub IL(λ) such that the stabilizer of λ in WJ is finite Then

wtM(λ J) =sumwisinWJ

weλprod

αisinπ(1minus eminusα)

Before proving Proposition 74 we note the similarity between the result and the followingformulation of the WeylndashKac character formula This is due to Atiyah and Bott for finitedimensional simple modules in finite type but appears to be new in this generality

Proposition 75 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

weλprod

αgt0(1minus eminusα)dim gα (76)

The right hand side of Equation (76) is expanded in the manner explained above Wenow prove Proposition 74 and then turn to Proposition 75

Proof of Proposition 74 By expanding wprodαisinπIJ (1 minus eminusα)minus1 note the right hand side

equals sumwisinWJ

summicroisinZgt0πIJ

weλminusmicroprod

αisinπJ (1minus eminusα)

Therefore we are done by the Integrable Slice Decomposition 41 and the existing result forintegrable highest weight modules with finite stabilizer [26]

Specialized to the trivial module in finite type we obtain the following curious consequencewhich roughly looks like a Weyl denominator formula without a choice of positive roots

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Corollary 77 Let g be of finite type with root system ∆ and let Π denote the set of all πwhere π is a simple system of roots for ∆ Then we haveprod

αisin∆

(1minus eα) =sumπisinΠ

prodβisinπ

(1minus eβ) (78)

Remark 79 When g is of finite type Theorem 71 follows from Proposition 24 and generalformulae for exponential sums over polyhedra due to Brion [8] for rational polytopes andLawrence [34] and KhovanskiindashPukhlikov [29] in general cf [3 Chapter 13] For regularweights one uses that the tangent cones are unimodular and hence the associated polyhedraare Delzant For general highest weights one may apply a deformation argument due toPostnikov [39] We thank Michel Brion for sharing this observation with us

Moreover it is likely that a similar approach works in infinite type though a cutoff argu-ment needs to be made owing to the fact that conv V is in general locally but not globallypolyhedral cf our companion work [11]

71 Some related results We conclude this section with several related observationswhich are not needed in the remainder of the paper beginning with Proposition 75

711 Parabolic AtiyahndashBott formula and the BernsteinndashGelfandndashGelfandndashLepowsky resolu-tion Proposition 75 may be deduced from the usual presentation of the parabolic WeylndashKaccharacter formula

Proposition 710 Fix λ isin hlowast J sub IL(λ) Then

chM(λ J) =sumwisinWJ

ew(λ+ρ)minusρprodαgt0(1minus eminusα)dim gα

(711)

To our knowledge even Proposition 710 does not appear in the literature in this generalityHowever via Levi induction one may reduce to the case of λ dominant integral J = I whereit is a famed result of Kumar [31 32] Along these lines we also record the BGGL resolutionfor arbitrary parabolic Verma modules which may be of independent interest

Proposition 712 Fix M(λ J) and J prime sub J Then M(λ J) admits a BGGL resolution iethere is an exact sequence

middot middot middot rarroplus

wisinWJprimeJ `(w)=2

M(w middot λ J prime)rarroplus

wisinWJprimeJ `(w)=1

M(w middot λ J prime)rarrM(λ J prime)rarrM(λ J)rarr 0

(713)

Here W J primeJ runs over the minimal length coset representatives of WJ primeWJ In particular we

have the WeylndashKac character formula

chM(λ J) =sum

wisinWJprimeJ

(minus1)`(w) chM(w middot λ J prime) (714)

This can be deduced by Levi induction from the result of Heckenberger and Kolb [17]which builds on [16 32 38]

712 Generalization of a result of Kass In the paper of Kass [26] that proves Theorem 71for integrable modules the main result is a recursive formula for the character of an integrablemodule After establishing notation we will state the analogous result for parabolic Vermamodules

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

18 GURBIR DHILLON AND APOORVA KHARE

For J sub I if ν isin hlowast lies in the J dot Tits cone write ν = wν middot ν for the unique J dotdominant weight in its WJ orbit For λ isin P+

J we write χλ = chM(λ J) and for ν lying inthe dot J Tits cone we set

χν =

(minus1)`(wν)χν if ν is in the J dominant chamber

0 otherwise

This makes sense because if ν lies in the J dominant chamber it is regular dominant underthe J dot action so there is a unique wν isinWJ such that wν middot ν = ν

Define the elements 〈Φ〉 isin Zgt0π and the integers c〈Φ〉 via the expansionprodαisin∆+π

(1minus eminusα)dim gα =sum〈Φ〉

c〈Φ〉eminus〈Φ〉

Informally this expansion counts the number of ways to write 〈Φ〉 as sums of non-simplepositive roots with consideration of the multiplicity of the root and the parity of the sum

For a parabolic Verma module M(λ J) by abuse of notation identify its weights with thecorresponding multiplicity-free character

wtM(λ J) =sum

microM(λJ)micro 6=0

emicro

With this notation we are ready to state the following

Proposition 715 Let M(λ J) be a parabolic Verma module such that the stabilizer of λ inWJ is finite Then

wtM(λ J) =sum〈Φ〉

c〈Φ〉χλminus〈Φ〉 (716)

Furthermore for all 〈Φ〉 we have (i) λminus 〈Φ〉 6 λ (ii) with equality if and only if Φ is empty

In Equation (716) we expand denominators as lsquohighest weightrsquo power series as explainedin Equation (73) The arguments of [26] or [40] apply with suitable modification to proveProposition 715

Remark 717 As the remainder of the paper concerns only symmetrizable g we now explainthe validity of earlier results for g (cf Section 32) should g and g differ The proofs in Sections4ndash7 prior to Remark 79 apply verbatim for g Alternatively recalling that the roots of g andg coincide [25 sect512] it follows from Equation (39) and the remarks following Proposition43 that the weights of parabolic Verma modules for g g coincide as do the weights of simplehighest weight modules Hence many of the previous results can be deduced directly for gfrom the case of g These arguments for g apply for any intermediate Lie algebra between gand g

8 A question of Brion on localization of D-modules and a geometricinterpretation of integrability

Throughout this section g is of finite type We will answer the question of Brion [9]discussed in Section 25 We first remind notation For λ a dominant integral weight considerLλ the line bundle on GB with H0(Lλ) L(λ) as g-modules For w isin W write Cw =BwBB for the Schubert cell and write Xw for its closure the Schubert variety

To affirmatively answer Brionrsquos question for any regular integral infinitesimal character itsuffices by translation to consider the regular block O0 Let V be a highest weight module

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

with highest weight w middot (minus2ρ) w isin W Write V for the corresponding regular holonomicD-module on GB and Vrh for the corresponding perverse sheaf under the RiemannndashHilbertcorrespondence Then VVrh are supported on Xw Note that for i isin IV Xsiw is a Schubertdivisor of Xw We will use D to denote the standard dualities on Category O regularholonomic D-modules and perverse sheaves which are intertwined by BeilinsonndashBernsteinlocalization and the RiemannndashHilbert correspondence

Theorem 81 For VV as above and J sub IV consider the union of Schubert divisorsZ = cupiisinIV JXsiw Write U = GB Z Then DH0(U DV) is of highest weight w middot (minus2ρ)and has integrability J

To prove Theorem 81 we will use the following geometric characterization of integrabilitywhich was promised in Theorem 12

Proposition 82 For J sub IL(λ) consider the smooth open subvariety of Xw given on com-plex points by

UJ = Cw t⊔iisinJ

Csiw (83)

Let VVrh be as above and set U = UIL(λ) Upon restricting to U we have

Vrh|U jCUIV [`(w)] (84)

In particular IV = i isin I the stalks of Vrh along Csiw are nonzero

Proof For J sub IL(λ) let PJ denote the corresponding parabolic subgroup of G ie with Lie

algebra lJ + n+ Then it is well known that the perverse sheaf corresponding to M(λ J) isjCPJwBB[`(w)]

The map M(λ IV ) rarr V rarr 0 yields a surjection on the corresponding perverse sheavesBy considering JordanndashHolder content it follows this map is an isomorphism when restrictedto UIL(λ)

as for y 6 w the only intersection cohomology sheaves ICy = ICXy which do notvanish upon restriction are ICw ICsiw i isin IL(λ) We finish by observing

jlowastUIL(λ)jCUIV [`(w)] jCUIL(λ)

capPIV wBB[`(w)] = jCUIV [`(w)]

We deduce the following D-module interpretation of integrability

Corollary 85 Let VU be as above The restriction of DV to U is jlowastOUIV If we define

IV = i isin I the fibers of DV are nonzero along Csiw then IV = IV Here we mean fibersin the sense of the underlying quasi-coherent sheaf of DV

Before proving Theorem 81 we first informally explain the idea Proposition 82 saysthat for i isin IL(wmiddotminus2ρ) the action of the corresponding sl2 rarr g on V is not integrable if andonly if Vrh has a lsquopolersquo on the Schubert divisor Xsiw Therefore to modify V so that it losesintegrability along Xsiw we will restrict Vrh to the complement and then extend by zero

Proof of Theorem 81 For ease of notation write X = Xw Write X = Z t U where Z isas above and GB = Z t U Note U cap Xw = U Write jU U rarr Xw jU U rarr GB forthe open embeddings and iZ Z rarr Xw i

primeZ Z rarr GB iXw Xw rarr GB for the closed

embeddingsWe will study H0(U V) by studying the behavior of V on U ie before pushing off of Xw

Formally write Vrh = iXlowastiXlowastVrh = iXlowastP where P is perverse The distinguished triangle

jU jUlowast rarr idrarr iZlowastiZ

lowast +1minusminusrarr (86)

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

20 GURBIR DHILLON AND APOORVA KHARE

gives the following exact sequence in perverse cohomology

0rarr pHminus1 iZlowastiZlowastP rarr pH0 jU jU

lowastP rarr P rarr pH0 iZlowastiZlowastP rarr 0

We now show in several steps that pH0 jU jUlowastP is a highest weight module with the desired

integrability

Step 1 pH0 jU jUlowastP is a highest weight module of highest weight w middot minus2ρ

By definition we have a surjection jCw CCw [`(w)] rarr P rarr 0 Since jCw CCw is supportedoff of the Schubert divisors we have jU jU

lowastCCw CCw By right exactness we obtainjCw CCw [`(w)]rarr pH0 jU jU

lowastP rarr 0 as desired We remark that similarly pH0 iZlowastiZlowastP = 0

Step 2 The integrability of pH0 jU jUlowastP is J In lsquoground to earthrsquo terms by Proposition

82 we need to look at this sheaf on U where by design it has the correct behavior Morecarefully we have

jlowastUpH0 jU jU

lowastP pH0 jlowastUjU jUlowastP

pH0 jUcapU jlowastUcapUP

pH0 jUcapU jlowastUcapUjUIV

C[`(w)]

pH0 jUIV capU C[`(w)] = jUIV capU

C[`(w)]

As UIV cap U = UJ we are done by Proposition 82We now push our analysis off of Xw To do so we use the isomorphism of distinguished

triangles

jU jUlowastiXlowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iprimeZ i

primeZlowastiXlowast

+1minusminusminusminusrarry y yiXlowastjU jU

lowast minusminusminusminusrarr iXlowast minusminusminusminusrarr iXlowastiZlowastiZlowast +1minusminusminusminusrarr

(87)

Here the middle vertical map is the identity and the left and right vertical isomorphismsare induced by adjunction Plainly the isomorphism (87) comes from an isomorphism ofshort exact sequences of functors for the abelian categories of sheaves of abelian groups

Translating our analysis of pH0 jU jUlowastP into the corresponding statement for g-modules

and using Equation (87) we obtain a surjection of highest weight modules

Γ(GBH0jU jUlowastV)rarr V rarr 0

where the former module has integrability J To finish the proof of Theorem 81 it remainsonly to identify the dual of Γ(GBH0jU

jUlowastV) with sections of DV on U But using standard

compatibilities of D and of composition of derived functors we obtain

DΓ(GBH0jU jUlowastV) Γ(GBDH0jU

jUlowastV)

Γ(GBH0DjU jlowastUV)

Γ(GBH0RjU lowastjUlowastDV)

R0Γ(GBRjU lowastjlowastUDV)

Γ(U DV)

If one thinks about the above proof in fact all we used about U (and U) was the inter-section of U with U The following proposition shows our U has the correct components incodimension gt 2 to mimic another feature of Brionrsquos example

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

Proposition 88 The construction of Theorem 81 sends parabolic Verma modules to par-abolic Verma modules

Proof For J sub K sub IL(wmiddotminus2ρ) we know that M(w middotminus2ρK) corresponds to jCPKwBB wherePK is the parabolic subgroup corresponding to K sub I Applying the construction (beforetaking perverse cohomology) we obtain jU j

lowastUjCPKwBB jCUcapPKwBB The claim follows

from the identity PKwBB Z = PJwBB ie the identity

WKw cupkisinKJy isinW y 6 skw = WJw

To see this identity recall by [7 Exercise 226 and proof of Proposition 244] that theassignment wkw 7rarr wkw is an isomorphism of posets WKw WK where w is the longestelement ofWK Multiplying the claimed identity on the right by wminus1 it is therefore equivalentto

WK cupkisinKJy isinWK y gt sk = WJ

which is clear

While the results in this section concern g of finite type we expect and would be interestedto see that similar results hold for g symmetrizable

9 Highest weight modules over symmetrizable quantum groups

We now extend many results of the previous sections to highest weight modules overquantum groups Uq(g) for g a KacndashMoody algebra Given a generalized Cartan matrixA as for g = g(A) to write down a presentation for the algebra Uq(g) via generators andexplicit relations one uses the symmetrizability of A When g is non-symmetrizable eventhe formulation of Uq(g) is subtle and is the subject of recent research [13] In light of thiswe restrict to Uq(g) where g is symmetrizable

91 Notation and preliminaries We begin by reminding standard definitions and no-tation Fix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonalmatrix D = diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) bea realization of A as before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6l 6 |I| minus rk(A) such that Por otimesZ C h and (βl αi) isin Z foralli isin I 1 6 l 6 |I| minus rk(A)Set P = λ isin hlowast (Por λ) sub Z to be the weight lattice We further retain the notationsρ b h lJ P

+J M(λ J) IL(λ) from previous sections note we may and do choose ρ isin P We

normalize the Killing form (middot middot) on hlowast to satisfy (αi αj) = diaij for all i j isin ILet q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is

a C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [18

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ

Tq rarr C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration

of αi i isin I We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering

on the set of weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0π We will mostly be

concerned with integral weights microq = qmicro for micro isin P which are defined via qmicro(qh) = q(hmicro) h isinPor

Given a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

22 GURBIR DHILLON AND APOORVA KHARE

Denote by wtV the set of weights microq Vmicroq 6= 0 A Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I For aweight microq let M(microq) L(microq) denote the Verma and simple Uq(g)-modules of highest weightmicroq respectively

Writing λq for the highest weight of V the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (91)

In this case the parabolic subgroup WIV acts on wtV by

si(λq)(qh) = λq(q

αi)minusαi(h)λq(qh) h isin Por

The braid relations can be checked using by specializing q to 1 cf [36] In particularw(qλ) = qwλ for w isinW and λ isin P

Given a weight λq and J sub IL(λq) the parabolic Verma module M(λq J) co-represents thefunctor

M m isinMλq eim = 0 foralli isin I fj acts nilpotently on mforallj isin J (92)

Note λq(qαj ) = plusmnqnj nj gt 0 forallj isin J cf [23 Proposition 23] Therefore

M(λq J) M(λq)(fnj+1j M(λq)λq forallj isin J) (93)

In what follows we will specialize highest weight modules at q = 1 as pioneered by Lusztig[36] Let A1 denote the local ring of rational functions f isin C(q) that are regular at q = 1 Fixan integral weight λq = qλ isin qP and a highest weight module V with highest weight vectorvλq isin Vλq Then the classical limit of V at q = 1 defined to be V 1 = A1 middot vλq(qminus 1)A1 middot vλq is a highest weight module over U(g) with highest weight λ Moreover the characters of Vand V 1 are ldquoequalrdquo ie upon identifying qP with P

92 A classification of highest weight modules to first order We begin by extendingTheorem 12 to quantum groups

Theorem 94 Let V be a highest weight module with integral highest weight λq The fol-lowing data are equivalent

(1) IV the integrability of V (2) conv V 1 the convex hull of the specialization of V (3) The stabilizer of chV in W

Proof The equivalence of the three statements follows from their classical counterparts usingthe equality of characters under specialization

93 A question of Bump and weight formulas for non-integrable simple modulesWe now extend the results of Section 5 to quantum groups

Theorem 95 Let λq be integral The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 95 we will need the Integrable Slice Decomposition for quantum par-abolic Verma modules

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 96 Let λq be an integral weight and M(λq J) = V a parabolic Verma moduleThen

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (97)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν In particular wtV 1 =wtM(λ1 J)

Proof It suffices to check the formula after specialization Since the characters of V and V 1

are ldquoequalrdquo IV = IV 1 It therefore suffices to check the surjection M(λ1 J) rarr V 1 inducesan equality of weights By the Integrable Slice Decomposition 41 it suffices to show that

qminusZgt0πIJλq sub wtM(λq J) But this is clear from (93) by considering weights and using

that fnj+1j M(λq)λq is a highest weight line for all j isin J

Remark 98 If g is of finite type then in fact V 1 M(λ1 J) The proof uses [6 Chapter 2]and the PBW theorem as well as arguments similar to the construction of Lusztigrsquos canonicalbasis [37] to define Uq(u

minusJ ) It would be interesting to know if the parabolic Verma module

M(λq J) specializes to M(λ1 J) for all symmetrizable g

Proof of Theorem 95 By Proposition 96 (1) implies (2) For the converse by the Integrable

Slice Decomposition (97) it suffices to show that qminusZgt0(ππIV )λq sub wtV This follows by

mimicking the proof of Theorem 58

As an application of Theorem 95 we obtain the following positive formulas for weights ofsimple modules wtL(λq)

Proposition 99 Suppose λq is integral Write l for the Levi subalgebra corresponding toIL(λq) and write Ll(νq) for the simple Uq(l) module with highest weight νq Then

wtL(λq) =⊔

microisinZgt0ππIL(λq)

wtLl(qminusmicroλq) (910)

Proposition 911 Let λq be integral and V = L(λq) Then

wtV 1 = (λ+ Zπ) cap conv V 1 (912)

By specialization this determines the weights of L(λq)

Proposition 913 Suppose λq is integral and has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wqν ν isin P+IL(λq)

qν 6 λq (914)

94 A question of Lepowsky on the weights of highest weight modules The resultand arguments of Section 6 apply without change to quantum groups

Theorem 915 Fix J sub IL(λq) and define Jpq = IL(λq) J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

24 GURBIR DHILLON AND APOORVA KHARE

95 The WeylndashKac formula for the weights of simple modules The main result ofSection 7 follows from combining Theorems 71 95 Proposition 96 and specializing

Theorem 916 For all integral λq with finite stabilizer in WIL(λq) we have

wtL(λq) =sum

wisinWIL(λq)

wλqprod

αisinπ(1minus qminusα) (917)

96 The case of non-integral weights We now explain how to extend the above resultsin this section to other highest weights We do so in two ways First we observe that withsome modifications Lusztigrsquos specialization method applies to any weight λq such that λq(q

h)is regular at q = 1 with value 1 forallh isin Por Calling such weights specializable one can showthat as before a highest weight module with specializable highest weight λq specializes to ahighest weight U(g)-module with highest weight λ1 isin hlowast given by

(h λ1) =λq(q

h)minus 1

q minus 1

∣∣∣∣q=1

h isin Por

Moreover the following holds

Theorem 918 The above results in this section all extend to λq specializable

Second we extend many of the above results to generic highest weights ie with finiteintegrable stabilizer In particular this covers all cases in finite and affine type the remainingcases in affine type being trivial modules As before we obtain

Theorem 919 Fix a weight λq and a highest weight module V such that the stabilizer ofits highest weight λq in WIV is finite The following are equivalent

(1) wtV = wtM(λq IV )

(2) wtIpVV = q

minusZgt0πIpV λq where IpV = IL(λq) IV

In particular if V is simple or more generally |IpV | 6 1 then wtV = wtM(λq IV )

To prove Theorem 919 we will need the Integrable Slice Decomposition for quantumparabolic Verma modules

Proposition 920 Let M(λq J) be a parabolic Verma module such that the stabilizer of λqin WJ is finite Then

wtM(λq J) =⊔

microisinZgt0(ππJ )

wtLlJ (qminusmicroλq) (921)

where LlJ (ν) denotes the simple Uq(lJ)-module of highest weight ν

Proof The inclusion sup follows from considering weights as in Proposition 41 The inclusionsub follows by using Proposition 43(2) which holds for quantum groups by specialization toq = 1

Proof of Theorem 919 By Proposition 920 (1) implies (2) For the converse by the In-

tegrable Slice Decomposition (921) it suffices to show that qminusZgt0(ππIV )λq sub wtV This

follows by mimicking the proof of Theorem 58

As an application of Theorem 919 Equation (910) holds on the nose for all λq with finitestabilizer in WIL(λq)

and Equation (914) holds rephrased as follows

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 922 Suppose λq has finite WIL(λq)-isotropy Then

wtL(λq) =⋃

wisinWIL(λq)

wνq νq 6 λq νq(qαi) = plusmnqni ni isin Zgt0 foralli isin IL(λq) (923)

Finally we extend the results of Section 6 to quantum groups as the same proof applies

Theorem 924 Fix λq and J sub IL(λq) such that the stabilizer of λq in WJ is finite and

define Jpq = IL(λq)J The following are equivalent

(1) wtV = wtM(λq J) for every V with IV = J (2) The Dynkin diagram for gJpq is complete

References

[1] Tomoyuki Arakawa and Peter Fiebig The linkage principle for restricted critical level representations ofaffine KacndashMoody algebras Compos Math 148(6)1787ndash1810 2012

[2] Michael F Atiyah and Raoul Bott A Lefschetz fixed point formula for elliptic complexes II ApplicationsAnn of Math (2) 88451ndash491 1968

[3] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics EuropeanMathematical Society (EMS) Zurich 2008

[4] Alexander Beilinson and Joseph Bernstein Localisation de g-modules C R Acad Sci Paris 292(1)15ndash18 1981

[5] Alexander Beilinson Victor Ginzburg and Wolfgang Soergel Koszul duality patterns in representationtheory J Amer Math Soc 9(2)473ndash527 1996

[6] Christopher P Bendel Daniel K Nakano Brian J Parshall and Cornelius Pillen Cohomology for quan-tum groups via the geometry of the nullcone Mem Amer Math Soc 229(1077)x+93 2014

[7] Anders Bjorner and Francesco Brenti Combinatorics of Coxeter groups volume 231 of Graduate Textsin Mathematics Springer New York 2005

[8] Michel Brion Polyedres et reseaux Enseign Math (2) 38(1-2)71ndash88 1992[9] Michel Brion Personal communication 2013

[10] Jean-Luc Brylinski and Masaki Kashiwara KazhdanndashLusztig conjecture and holonomic systems In-vent Math 64(3)387ndash410 1981

[11] Gurbir Dhillon and Apoorva Khare Faces of highest weight modules and the universal Weyl polyhedronPreprint arXiv161100114 2016

[12] Gurbir Dhillon and Apoorva Khare The WeylndashKac weight formula Sem Lothar Combin in press 2017[13] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[14] Edward Frenkel Langlands correspondence for loop groups volume 103 of Cambridge Studies in Advanced

Mathematics Cambridge University Press Cambridge 2007[15] Edward Frenkel and Dennis Gaitsgory Localization of g-modules on the affine Grassmannian Ann of

Math (2) 170(3)1339ndash1381 2009[16] Howard Garland and James Lepowsky Lie algebra homology and the MacdonaldndashKac formulas In-

vent Math 34(1)37ndash76 1976[17] Istvan Heckenberger and Stefan Kolb On the BernsteinndashGelfandndashGelfand resolution for KacndashMoody

algebras and quantized enveloping algebras Transform Groups 12(4)647ndash655 2007[18] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[19] Ronald S Irving Singular blocks of the category O Math Z 204(2)209ndash224 1990[20] James E Humphreys Representations of semisimple Lie algebras in the BGG category O volume 94 of

Graduate Studies in Mathematics American Mathematical Society Providence RI 2008[21] Jens C Jantzen Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren Math

Ann 226(1)53ndash65 1977[22] Jens C Jantzen Moduln mit einem hochsten Gewicht volume 750 of Lecture Notes in Mathematics

Springer Berlin 1979[23] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996

26 GURBIR DHILLON AND APOORVA KHARE

[24] Victor G Kac Infinite-dimensional Lie algebras and the Dedekind η-function (Russian) Funkt analys yego prilozh 8(1)77ndash78 1974 (English translation in Funct Anal Appl 8(1)68ndash70 1974)

[25] Victor G Kac Infinite-dimensional Lie algebras Cambridge University Press Cambridge third edition1990

[26] Steven N Kass A recursive formula for characters of simple Lie algebras J Algebra 137(1)126ndash1441991

[27] David A Kazhdan and George Lusztig Representations of Coxeter groups and Hecke algebras In-vent Math 53(2)165ndash184 1979

[28] Apoorva Khare Faces and maximizer subsets of highest weight modules J Algebra 45532ndash76 2016[29] Askold G Khovanskii and Alexander V Pukhlikov Finitely additive measures of virtual polyhedra In

St Petersburg Mathematical Journal 4(2)337ndash356 1993[30] Bertram Kostant Lie algebra cohomology and the generalized BorelndashWeil theorem Ann of Math (2)

74329ndash387 1961[31] Shrawan Kumar Demazure character formula in arbitrary KacndashMoody setting Invent Math 89(2)395ndash

423 1987[32] Shrawan Kumar BernsteinndashGelprimefandndashGelprimefand resolution for arbitrary KacndashMoody algebras Math Ann

286(4)709ndash729 1990[33] Shrawan Kumar KacndashMoody groups their flag varieties and representation theory volume 204 of Progress

in Mathematics Birkhauser Boston Inc Boston MA 2002[34] Jim Lawrence Rational-function-valued valuations on polyhedra Discrete and computational geometry

(New Brunswick NJ 19891990) pp 199ndash208 DIMACS Series in Discrete Mathematics and TheoreticalComputer Science 6 American Mathematical Society Providence 1991

[35] James Lepowsky A generalization of the BernsteinndashGelfandndashGelfand resolution J Algebra 49(2)496ndash511 1977

[36] George Lusztig Quantum deformations of certain simple modules over enveloping algebras Adv Math70(2)237ndash249 1988

[37] George Lusztig Canonical bases arising from quantized enveloping algebras J Amer Math Soc3(2)447ndash498 1990

[38] Michael Murray and John Rice A geometric realisation of the Lepowsky Bernstein Gelprimefand Gelprimefandresolution Proc Amer Math Soc 114(2)553ndash559 1992

[39] Alexander Postnikov Permutohedra associahedra and beyond Int Math Res Not IMRN (6)1026ndash1106 2009

[40] Waldeck Schutzer A new character formula for Lie algebras and Lie groups J Lie Theory 22(3)817ndash8382012

[41] Wolfgang Soergel Kategorie O perverse Garben und Moduln uber den Koinvarianten zur WeylgruppeJ Amer Math Soc 3(2)421ndash445 1990

[42] Mark A Walton Polytope sums and Lie characters In Symmetry in physics volume 34 of CRM ProcLecture Notes pages 203ndash214 Amer Math Soc Providence RI 2004

[43] Hermann Weyl ldquoTheorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transfor-mationen I II IIIrdquo Math Z 23 pp 271ndash309 24 pp 328ndash376 24 pp 377ndash395 1925ndash1926

Department of Mathematics Stanford University Stanford CA 94305 USAE-mail address GD gsdstanfordedu AK kharestanfordedu