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representation theory and geometry
Geordie Williamson
University of Sydneyhttp://www.maths.usyd.edu.au/u/geordie/ICM.pdf
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representations
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representations
⇝ Sym
¨
˚
˝
˛
‹
‚
Ă GLpR3q.
We obtain a representation of our group of symmetries
ρ : G Ñ GLpVq.
2
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representations
⇝ Sym
¨
˚
˝
˛
‹
‚
Ă GLpR3q.
We obtain a representation of our group of symmetries
ρ : G Ñ GLpVq.
2
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representations
e1
e2
´e1 ´ e2
#˜
1 00 1
¸
,
˜
1 ´10 ´1
¸
,
˜
´1 0´1 1
¸
,
˜
0 ´11 ´1
¸
,
˜
´1 1´1 0
¸
,
˜
0 11 0
¸+
3
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why study representations?
Symmetric group SnThe problem of understanding
tSn-setsu{isomorphism „Ø tsubgroups of Snu{conjugation
is hard. The theory of representations of Sn is rich, highly-developedand useful.
Galois representationsThe passage(s)
tvarieties{Qu ÝÑ tGalois representationsu
is one of the most powerful tools of modern number theory.
4
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why study representations?
Symmetric group SnThe problem of understanding
tSn-setsu{isomorphism „Ø tsubgroups of Snu{conjugation
is hard.
The theory of representations of Sn is rich, highly-developedand useful.
Galois representationsThe passage(s)
tvarieties{Qu ÝÑ tGalois representationsu
is one of the most powerful tools of modern number theory.
4
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why study representations?
Symmetric group SnThe problem of understanding
tSn-setsu{isomorphism „Ø tsubgroups of Snu{conjugation
is hard. The theory of representations of Sn is rich, highly-developedand useful.
Galois representationsThe passage(s)
tvarieties{Qu ÝÑ tGalois representationsu
is one of the most powerful tools of modern number theory.
4
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why study representations?
Symmetric group SnThe problem of understanding
tSn-setsu{isomorphism „Ø tsubgroups of Snu{conjugation
is hard. The theory of representations of Sn is rich, highly-developedand useful.
Galois representationsThe passage(s)
tvarieties{Qu ÝÑ tGalois representationsu
is one of the most powerful tools of modern number theory.
4
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simple representations
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on R3.
There are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
R3 “ L‘ H “ trivial ‘
A representation V of a group G is simple or irreducible if its onlyG-invariant subspaces are t0u and V.
A representation is semi-simple if it is isomorphic to a direct sum ofsimple representations.
5
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simple representations
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on R3. There are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
R3 “ L‘ H “ trivial ‘
A representation V of a group G is simple or irreducible if its onlyG-invariant subspaces are t0u and V.
A representation is semi-simple if it is isomorphic to a direct sum ofsimple representations.
5
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simple representations
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on R3. There are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
R3 “ L‘ H “ trivial ‘
A representation V of a group G is simple or irreducible if its onlyG-invariant subspaces are t0u and V.
A representation is semi-simple if it is isomorphic to a direct sum ofsimple representations.
5
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simple representations
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on R3. There are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
R3 “ L‘ H “ trivial ‘
A representation V of a group G is simple or irreducible if its onlyG-invariant subspaces are t0u and V.
A representation is semi-simple if it is isomorphic to a direct sum ofsimple representations.
5
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simple representations
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on R3. There are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
R3 “ L‘ H “ trivial ‘
A representation V of a group G is simple or irreducible if its onlyG-invariant subspaces are t0u and V.
A representation is semi-simple if it is isomorphic to a direct sum ofsimple representations.
5
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jordan-hölder theorem
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on F33. (Here F3 “ Z{3Z is the finite field with 3 elements.)
As before, there are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
However now L Ă H because 3 “ 0. We obtain a composition series
0 Ă L Ă H Ă F33.
We write (“Grothendieck group”, “multiplicities”)
rF33s “ rLs ` rH{Ls ` rF33{Hs “ 2rtrivials ` rsigns.
trivial
sign
trivial
6
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jordan-hölder theorem
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on F33. (Here F3 “ Z{3Z is the finite field with 3 elements.)As before, there are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
However now L Ă H because 3 “ 0. We obtain a composition series
0 Ă L Ă H Ă F33.
We write (“Grothendieck group”, “multiplicities”)
rF33s “ rLs ` rH{Ls ` rF33{Hs “ 2rtrivials ` rsigns.
trivial
sign
trivial
6
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jordan-hölder theorem
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on F33. (Here F3 “ Z{3Z is the finite field with 3 elements.)As before, there are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
However now L Ă H because 3 “ 0.
We obtain a composition series
0 Ă L Ă H Ă F33.
We write (“Grothendieck group”, “multiplicities”)
rF33s “ rLs ` rH{Ls ` rF33{Hs “ 2rtrivials ` rsigns.
trivial
sign
trivial
6
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jordan-hölder theorem
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on F33. (Here F3 “ Z{3Z is the finite field with 3 elements.)As before, there are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
However now L Ă H because 3 “ 0. We obtain a composition series
0 Ă L Ă H Ă F33.
We write (“Grothendieck group”, “multiplicities”)
rF33s “ rLs ` rH{Ls ` rF33{Hs “ 2rtrivials ` rsigns.
trivial
sign
trivial
6
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jordan-hölder theorem
ExampleConsider the symmetric group S3. It acts via permutation ofcoordinates on F33. (Here F3 “ Z{3Z is the finite field with 3 elements.)As before, there are two invariant subspaces:
L “ tall coordinates equalu, H “ tcoordinates sum to zerou.
However now L Ă H because 3 “ 0. We obtain a composition series
0 Ă L Ă H Ă F33.
We write (“Grothendieck group”, “multiplicities”)
rF33s “ rLs ` rH{Ls ` rF33{Hs “ 2rtrivials ` rsigns.
trivial
sign
trivial
6
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simple representations
representations Ø “matter”
simple representations Ø “elements”␣
, ,(
semi-simple Ø “elements don’t interract”
We search for a classification (“periodic table”), character formulas(“mass”, “number of neutrons”), …
7
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simple representations
representations Ø “matter”
simple representations Ø “elements”␣
, ,(
semi-simple Ø “elements don’t interract”
We search for a classification (“periodic table”), character formulas(“mass”, “number of neutrons”), …
7
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plan of talk
“Semi-simple world”‚ Finite groups:Maschke’s theorem (1897).
‚ Compact Lie groups:Weyl’s theorem (1925).
“Beyond the semi-simple world”‚ Infinite-dimensional representations of Lie algebras:Jantzen conjecture and Kazhdan-Lusztig conjecture (1979).
‚ Modular representations of reductive algebraic groups:Lusztig conjecture (1980) and new character formula (2018).
‚ Modular representations of symmetric groups:Billiards conjecture (2017).
Related situations: non-compact Lie groups, p-adic groups…
8
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plan of talk
“Semi-simple world”‚ Finite groups:Maschke’s theorem (1897).
‚ Compact Lie groups:Weyl’s theorem (1925).
“Beyond the semi-simple world”‚ Infinite-dimensional representations of Lie algebras:Jantzen conjecture and Kazhdan-Lusztig conjecture (1979).
‚ Modular representations of reductive algebraic groups:Lusztig conjecture (1980) and new character formula (2018).
‚ Modular representations of symmetric groups:Billiards conjecture (2017).
Related situations: non-compact Lie groups, p-adic groups…
8
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plan of talk
“Semi-simple world”‚ Finite groups:Maschke’s theorem (1897).
‚ Compact Lie groups:Weyl’s theorem (1925).
“Beyond the semi-simple world”‚ Infinite-dimensional representations of Lie algebras:Jantzen conjecture and Kazhdan-Lusztig conjecture (1979).
‚ Modular representations of reductive algebraic groups:Lusztig conjecture (1980) and new character formula (2018).
‚ Modular representations of symmetric groups:Billiards conjecture (2017).
Related situations: non-compact Lie groups, p-adic groups…
8
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plan of talk
“Semi-simple world”‚ Finite groups:Maschke’s theorem (1897).
‚ Compact Lie groups:Weyl’s theorem (1925).
“Beyond the semi-simple world”‚ Infinite-dimensional representations of Lie algebras:Jantzen conjecture and Kazhdan-Lusztig conjecture (1979).
‚ Modular representations of reductive algebraic groups:Lusztig conjecture (1980) and new character formula (2018).
‚ Modular representations of symmetric groups:Billiards conjecture (2017).
Related situations: non-compact Lie groups, p-adic groups…
8
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plan of talk
“Semi-simple world”‚ Finite groups:Maschke’s theorem (1897).
‚ Compact Lie groups:Weyl’s theorem (1925).
“Beyond the semi-simple world”‚ Infinite-dimensional representations of Lie algebras:Jantzen conjecture and Kazhdan-Lusztig conjecture (1979).
‚ Modular representations of reductive algebraic groups:Lusztig conjecture (1980) and new character formula (2018).
‚ Modular representations of symmetric groups:Billiards conjecture (2017).
Related situations: non-compact Lie groups, p-adic groups…
8
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plan of talk
“Semi-simple world”‚ Finite groups:Maschke’s theorem (1897).
‚ Compact Lie groups:Weyl’s theorem (1925).
“Beyond the semi-simple world”‚ Infinite-dimensional representations of Lie algebras:Jantzen conjecture and Kazhdan-Lusztig conjecture (1979).
‚ Modular representations of reductive algebraic groups:Lusztig conjecture (1980) and new character formula (2018).
‚ Modular representations of symmetric groups:Billiards conjecture (2017).
Related situations: non-compact Lie groups, p-adic groups…8
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common features
representation theory geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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common features
representation theory
geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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common features
representation theory geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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common features
representation theory geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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common features
representation theory
geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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common features
representation theory
geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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common features
representation theory
geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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common features
representation theory
geometry
invariant formssymmetric, hermitian, …
geometry
Related feature:(hidden) semi-simplicity
geometric structure
A geometric structure on a real (resp. complex) vector space V willmean a non-degenerate symmetric (resp. Hermitian) form on V.
We do not assume that our forms are positive definite; signature plays animportant role throughout.
9
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the semi-simple world
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maschke’s theorem
Maschke (1897)Any representation V of a finite group G over R or C is semi-simple.
Observation 1: If V has a positive-definite G-invariant geometricstructure, then V is semi-simple.
If U Ă V is a subrepresentation, then V “ U‘ UK.
Observation 2: Any representation of G admits a positive-definitegeometric structure.
Take a positive-definite geometric structure x´,´y on V. Then
xv,wyG :“1
|G|
ÿ
xgv,gwy
defines a positive-definite and G-invariant geometric structure.
11
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maschke’s theorem
Maschke (1897)Any representation V of a finite group G over R or C is semi-simple.
Observation 1: If V has a positive-definite G-invariant geometricstructure, then V is semi-simple.
If U Ă V is a subrepresentation, then V “ U‘ UK.
Observation 2: Any representation of G admits a positive-definitegeometric structure.
Take a positive-definite geometric structure x´,´y on V. Then
xv,wyG :“1
|G|
ÿ
xgv,gwy
defines a positive-definite and G-invariant geometric structure.
11
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maschke’s theorem
Maschke (1897)Any representation V of a finite group G over R or C is semi-simple.
Observation 1: If V has a positive-definite G-invariant geometricstructure, then V is semi-simple.
If U Ă V is a subrepresentation, then V “ U‘ UK.
Observation 2: Any representation of G admits a positive-definitegeometric structure.
Take a positive-definite geometric structure x´,´y on V. Then
xv,wyG :“1
|G|
ÿ
xgv,gwy
defines a positive-definite and G-invariant geometric structure.
11
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geometric structure
Example of “semi-simplicity via introduction of geometric structure”.
If V is simple and defined over the complex numbers, then Schur’slemma shows that the geometric structure is unique up to positivescalar.
This is an example of “unicity of geometric structure”.
⇝
12
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geometric structure
Example of “semi-simplicity via introduction of geometric structure”.
If V is simple and defined over the complex numbers, then Schur’slemma shows that the geometric structure is unique up to positivescalar.
This is an example of “unicity of geometric structure”.
⇝
12
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geometric structure
Example of “semi-simplicity via introduction of geometric structure”.
If V is simple and defined over the complex numbers, then Schur’slemma shows that the geometric structure is unique up to positivescalar.
This is an example of “unicity of geometric structure”.
⇝
12
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weyl’s theorem
Consider a compact Lie group K, e.g. S1 or SU2 or a finite group.
Weyl generalised these observations to K, with sum replaced byintegral:
xv,wyK :“
ż
Kxgv,gwydµ
Weyl (1925)
Any continuous representation of a compact Lie group K issemi-simple.
Existence and uniqueness of geometric structure still holds.
13
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weyl’s theorem
Consider a compact Lie group K, e.g. S1 or SU2 or a finite group.
Weyl generalised these observations to K, with sum replaced byintegral:
xv,wyK :“
ż
Kxgv,gwydµ
Weyl (1925)
Any continuous representation of a compact Lie group K issemi-simple.
Existence and uniqueness of geometric structure still holds.
13
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weyl’s theorem
Consider a compact Lie group K, e.g. S1 or SU2 or a finite group.
Weyl generalised these observations to K, with sum replaced byintegral:
xv,wyK :“
ż
Kxgv,gwydµ
Weyl (1925)
Any continuous representation of a compact Lie group K issemi-simple.
Existence and uniqueness of geometric structure still holds.
13
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cartan’s letter
Élie Cartan to Hermann Weyl, 28 of March 1925:
“…the difficulty, I dare not say the impossibility, of finding a proofwhich does not leave the strictly infinitesimal domain shows thenecessity of not sacrificing either point of view …”An algebraic (“ infinitesimal”) proof took 10 years, and involves the Casimir element (arises from an invariant form called the trace form).
14
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extended example: su2 and sl2
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su2
SU2 “
#
A “
˜
a bc d
¸ ˇ
ˇ
ˇ
ˇ
ˇ
AA˚ “ id,detA “ 1+
“unit
quaternions.
LiepSU2qC “ sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
“I don’t think it is the representations themselves, but the groups. I findSU2, SL2, Sn etc. amazing and beautiful animals (if I have a favourite, itis SU2), but will probably never really understand them. I mightsomeday understand their linear shadows though...”
– Quindici
16
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su2
SU2 “
#
A “
˜
a bc d
¸ ˇ
ˇ
ˇ
ˇ
ˇ
AA˚ “ id,detA “ 1+
“unit
quaternions.
LiepSU2qC “ sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
“I don’t think it is the representations themselves, but the groups. I findSU2, SL2, Sn etc. amazing and beautiful animals (if I have a favourite, itis SU2), but will probably never really understand them. I mightsomeday understand their linear shadows though...”
– Quindici
16
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simple representations of su2
SU2 acts on its “natural representation”:
C2 “ C
˜
01
¸
‘ C
˜
10
¸
“ CY‘ CX.
For any m ě 0, SU2 acts naturally on homogenous polynomials in X, Yof degree m:
Lm :“ CYm ‘ CYm´1X‘ ¨ ¨ ¨ ‘ CYmXm´1 ‘ CXm.
The Lm for m ě 0 are all irreducible representations of SU2.
“spherical harmonics”, “quantum mechanics”.
17
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action of the lie algebra
Differentiate to get representation of the (complexified) Lie algebra
LiepSU2qC “ sl2pCq
sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
Action on Lm (here m “ 5):
e
f
h
C
Y5
C
Y4X1
C
Y3X2
C
Y2X3
C
YX4
C
X5
012345
1 2 3 4 5
531´1´3´5
18
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action of the lie algebra
Differentiate to get representation of the (complexified) Lie algebra
LiepSU2qC “ sl2pCq
sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
Action on Lm (here m “ 5):
e
f
h
CY5 CY4X1 CY3X2 CY2X3 CYX4 CX5
012345
1 2 3 4 5
531´1´3´5
18
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action of the lie algebra
Differentiate to get representation of the (complexified) Lie algebra
LiepSU2qC “ sl2pCq
sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
Action on Lm (here m “ 5):
e
f
h
C
Y5
C
Y4X1
C
Y3X2
C
Y2X3
C
YX4
C
X5
012345
1 2 3 4 5
531´1´3´5
18
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action of the lie algebra
Differentiate to get representation of the (complexified) Lie algebra
LiepSU2qC “ sl2pCq
sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
Action on Lm (here m “ 5):
e
f
h
C
Y5
C
Y4X1
C
Y3X2
C
Y2X3
C
YX4
C
X5
012345
1 2 3 4 5
531´1´3´5
18
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action of the lie algebra
Differentiate to get representation of the (complexified) Lie algebra
LiepSU2qC “ sl2pCq
sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
Action on Lm (here m “ 5):
e
f
h
C
Y5
C
Y4X1
C
Y3X2
C
Y2X3
C
YX4
C
X50
12345
1 2 3 4 5
531´1´3´5
18
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verma modules
For SU2 and sl2pCq we can do everything “by hand”.
This is no longer possible for more complicated groups.
In the algebraic theory an important role is played byinfinite-dimensional representations called Verma modules.
The study of Verma modules has led to important advances inrepresentation theory beyond the semi-simple world.
For sl2pCq they depend on a parameter λ P C.
As vector spaces:∆λ “
à
iPZě0
Cvi
19
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verma modules
For SU2 and sl2pCq we can do everything “by hand”.
This is no longer possible for more complicated groups.
In the algebraic theory an important role is played byinfinite-dimensional representations called Verma modules.
The study of Verma modules has led to important advances inrepresentation theory beyond the semi-simple world.
For sl2pCq they depend on a parameter λ P C.
As vector spaces:∆λ “
à
iPZě0
Cvi
19
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verma modules
For SU2 and sl2pCq we can do everything “by hand”.
This is no longer possible for more complicated groups.
In the algebraic theory an important role is played byinfinite-dimensional representations called Verma modules.
The study of Verma modules has led to important advances inrepresentation theory beyond the semi-simple world.
For sl2pCq they depend on a parameter λ P C.
As vector spaces:∆λ “
à
iPZě0
Cvi
19
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verma modules
For SU2 and sl2pCq we can do everything “by hand”.
This is no longer possible for more complicated groups.
In the algebraic theory an important role is played byinfinite-dimensional representations called Verma modules.
The study of Verma modules has led to important advances inrepresentation theory beyond the semi-simple world.
For sl2pCq they depend on a parameter λ P C.
As vector spaces:∆λ “
à
iPZě0
Cvi
19
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verma modules
sl2pCq “C
˜
0 01 0
¸
‘ C
˜
1 00 ´1
¸
‘ C
˜
0 10 0
¸
f h e
“ “ “
The Verma module ∆λ determined by λ P C:
e
f
h
. . . Cv5 Cv4 Cv3 Cv2 Cv1 Cv00
λλ ´ 1λ ´ 2λ ´ 3λ ´ 4λ ´ 5
6 5 4 3 2 1
λλ ´ 2λ ´ 4λ ´ 6λ ´ 8λ ´ 10
20
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structure of verma modules
e
f
h
. . . Cv5 Cv4 Cv3 Cv2 Cv1 Cv00
λλ ´ 1λ ´ 2λ ´ 3λ ´ 4λ ´ 5
6 5 4 3 2 1
λλ ´ 2λ ´ 4λ ´ 6λ ´ 8λ ´ 10
λ ‰ 0, 1, 2, 3, . . . : ∆λ is simple, call it Lλ. λ
λ “ 0, 1, 2, 3, . . . : ∆λ is not simple. ?
?
?
21
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structure of verma modules
Example λ “ 2
e
f
h
. . . ‚ ‚ ‚ ‚ ‚ ‚
0210´1´2´3
6 5 4 3 2 1
20´2´4´6´8
We have a subrepresentation isomorphic to ∆´4, and
? ∆2{∆´4 – L2 ´42
(L2 is our simple finite-dimensional representation from earlier.) 22
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structure of verma modules
λ ‰ 0, 1, 2, 3, . . . : ∆λ is simple and infinite-dimensional. λ
λ “ 0, 1, 2, 3, . . . : ∆λ is not simple. ´λ ´ 2λ
Summary:
(a) A single family of representations (Verma modules) produces allsimple finite-dimensional representations.
(b) We get new infinite-dimensional simple representations.(c) The structure of Verma modules varies (subtly) based on the
parameter.
23
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structure of verma modules
λ ‰ 0, 1, 2, 3, . . . : ∆λ is simple and infinite-dimensional. λ
λ “ 0, 1, 2, 3, . . . : ∆λ is not simple. ´λ ´ 2λ
Summary:
(a) A single family of representations (Verma modules) produces allsimple finite-dimensional representations.
(b) We get new infinite-dimensional simple representations.(c) The structure of Verma modules varies (subtly) based on the
parameter.
23
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structure of verma modules
λ ‰ 0, 1, 2, 3, . . . : ∆λ is simple and infinite-dimensional. λ
λ “ 0, 1, 2, 3, . . . : ∆λ is not simple. ´λ ´ 2λ
Summary:(a) A single family of representations (Verma modules) produces all
simple finite-dimensional representations.
(b) We get new infinite-dimensional simple representations.(c) The structure of Verma modules varies (subtly) based on the
parameter.
23
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structure of verma modules
λ ‰ 0, 1, 2, 3, . . . : ∆λ is simple and infinite-dimensional. λ
λ “ 0, 1, 2, 3, . . . : ∆λ is not simple. ´λ ´ 2λ
Summary:(a) A single family of representations (Verma modules) produces all
simple finite-dimensional representations.(b) We get new infinite-dimensional simple representations.
(c) The structure of Verma modules varies (subtly) based on theparameter.
23
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structure of verma modules
λ ‰ 0, 1, 2, 3, . . . : ∆λ is simple and infinite-dimensional. λ
λ “ 0, 1, 2, 3, . . . : ∆λ is not simple. ´λ ´ 2λ
Summary:(a) A single family of representations (Verma modules) produces all
simple finite-dimensional representations.(b) We get new infinite-dimensional simple representations.(c) The structure of Verma modules varies (subtly) based on the
parameter.
23
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kazhdan-lusztig conjecture
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the weyl group
g is a complex semi-simple Lie algebra.
h Ă g a Cartan subalgebra.
W the Weyl group, which acts on h as a reflection group.
Example
g “ slnpCq “ t nˆ n matrices X | trX “ 0u.
h “ diagonal matrices Ă slnpCq
W “ Sn acting on h via permutations.
Motivation
We think of the finite group W as being the skeleton of g.
We try to answer questions about g in terms of W.
25
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the weyl group
g is a complex semi-simple Lie algebra.
h Ă g a Cartan subalgebra.
W the Weyl group, which acts on h as a reflection group.
Example
g “ slnpCq “ t nˆ n matrices X | trX “ 0u.
h “ diagonal matrices Ă slnpCq
W “ Sn acting on h via permutations.
Motivation
We think of the finite group W as being the skeleton of g.
We try to answer questions about g in terms of W.
25
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verma modules revisited
g is a complex semi-simple Lie algebra.
“weight” λ P h˚ ⇝ Verma module ∆λ.
∆λ has unique simple quotient ∆λ ↠ LλLλ is called a simple highest weight module.
Example sl2pCq
If λ ‰ 0, 1, . . . , Lλ “ ∆λ is infinite dimensional.If λ “ 0, 1, . . . then Lλ is finite dimensional.
Basic problem
Describe the structure of ∆λ.Which simple modules occur with which multiplicity?
26
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verma modules revisited
g is a complex semi-simple Lie algebra.
“weight” λ P h˚ ⇝ Verma module ∆λ.
∆λ has unique simple quotient ∆λ ↠ LλLλ is called a simple highest weight module.
Example sl2pCq
If λ ‰ 0, 1, . . . , Lλ “ ∆λ is infinite dimensional.If λ “ 0, 1, . . . then Lλ is finite dimensional.
Basic problem
Describe the structure of ∆λ.Which simple modules occur with which multiplicity?
26
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verma modules revisited
g is a complex semi-simple Lie algebra.
“weight” λ P h˚ ⇝ Verma module ∆λ.
∆λ has unique simple quotient ∆λ ↠ LλLλ is called a simple highest weight module.
Example sl2pCq
If λ ‰ 0, 1, . . . , Lλ “ ∆λ is infinite dimensional.If λ “ 0, 1, . . . then Lλ is finite dimensional.
Basic problem
Describe the structure of ∆λ.Which simple modules occur with which multiplicity?
26
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verma modules revisited
g is a complex semi-simple Lie algebra.
“weight” λ P h˚ ⇝ Verma module ∆λ.
∆λ has unique simple quotient ∆λ ↠ LλLλ is called a simple highest weight module.
Example sl2pCq
If λ ‰ 0, 1, . . . , Lλ “ ∆λ is infinite dimensional.If λ “ 0, 1, . . . then Lλ is finite dimensional.
Basic problem
Describe the structure of ∆λ.Which simple modules occur with which multiplicity?
26
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kazhdan-lusztig conjecture
∆λ : Verma module. Lλ : simple highest weight module.
Kazhdan-Lusztig conjecture (1979)
r∆λs “ÿ
µ
Pλ,µp1qrLµs.
Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.
(a) New paradigm in representation theory.(b) Pλ,µ only depends on a pair of elements the Weyl group W of g.
27
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kazhdan-lusztig conjecture
∆λ : Verma module. Lλ : simple highest weight module.
Kazhdan-Lusztig conjecture (1979)
r∆λs “ÿ
µ
Pλ,µp1qrLµs.
Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.
(a) New paradigm in representation theory.
(b) Pλ,µ only depends on a pair of elements the Weyl group W of g.
27
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kazhdan-lusztig conjecture
∆λ : Verma module. Lλ : simple highest weight module.
Kazhdan-Lusztig conjecture (1979)
r∆λs “ÿ
µ
Pλ,µp1qrLµs.
Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.
(a) New paradigm in representation theory.(b) Pλ,µ only depends on a pair of elements the Weyl group W of g.
27
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kazhdan-lusztig conjecture
∆λ : Verma module. Lλ : simple highest weight module.
Kazhdan-Lusztig conjecture (1979)
r∆x¨0s “ÿ
yPWPx¨0,y¨0p1qrLy¨0s.
Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.
(a) New paradigm in representation theory.(b) Pλ,µ only depends on a pair of elements the Weyl group W of g.
28
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29
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1
130
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v1
1
131
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v2
v1
v1
1
132
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v3
v2
v2
v1
v1
1
133
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v4v3
v3
v3
v2
v2
v2
v2
v1
v1
v1
1
134
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v5v4
v4
v4
v3
v3
v3
v3
v3
v2 v2
v2
v2
v2
v1
v1
v1
1
135
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v4 ` v6v5
v3 ` v5
v3 ` v5
v4
v4
v4
v2 ` v4
v4
v2 ` v4
v3
v3
v3
v3
v3
v1 ` v3
v3
v3
v2 v2
v2
v2
v2
v2
v2
v1
v1
v1
v1
1
136
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v5 ` v7v6
v4 ` v6
v4 ` v6
v5
v5
v5
v3 ` v5
v5
v3 ` v5
v4
v4
v4
v4
v4
v4
v2 ` v4
v4
v4
v3
v3
v3
v3
v3
v3
v3
v3
v3
v3
v2
v2
v2
v2
v2
v2
v2
v2
v1
v1
v1
v1
1
137
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2v6 ` v82v5 ` v7
2v5 ` v7
2v5 ` v7
v4 ` v6
v4 ` v6
2v4 ` v6
2v4 ` v6
2v4 ` v6
2v4 ` v6
v3 ` v5
v5
v3 ` v5
v5
v3 ` v5
2v3 ` v5
2v3 ` v5
v3 ` v5
2v3 ` v5
v4
v4
v4
v4
v4
v2 ` v4
v4
v2 ` v4
2v2 ` v4
v2 ` v4
v4
v2 ` v4
v3
v3
v3
v3
v3
v3
v3
v1 ` v3
v3
v1 ` v3
v3
v3
v3
v2
v2
v2
v2
v2
v2
v2
v2
v2
v2
v1
v1
v1
v1
v1
1
138
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potted history
Geometric proof (’80s)
Algebraic proof (’10s)
Kazhdan-Lusztig conjecture Brylinsky-Kashiwara
Elias-W. 2014,
(multiplicity “ Py,xp1qq Beilinson-Bernstein
followingSoergel 1990
Jantzen conjecture Beilinson-Bernstein W. 2016 following(graded multiplicity “ Py,xpvq) Soergel 2008,
Kübel 2012
Geometric proofs: D-modules, perverse sheaves, weights…
Algebraic proofs: “shadows of Hodge theory”,i.e. invariant forms (“geometric structures”)still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann
39
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potted history
Geometric proof (’80s)
Algebraic proof (’10s)
Kazhdan-Lusztig conjecture Brylinsky-Kashiwara
Elias-W. 2014,
(multiplicity “ Py,xp1qq Beilinson-Bernstein
followingSoergel 1990
Jantzen conjecture Beilinson-Bernstein
W. 2016 following
(graded multiplicity “ Py,xpvq)
Soergel 2008,Kübel 2012
Geometric proofs: D-modules, perverse sheaves, weights…
Algebraic proofs: “shadows of Hodge theory”,i.e. invariant forms (“geometric structures”)still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann
39
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potted history
Geometric proof (’80s) Algebraic proof (’10s)
Kazhdan-Lusztig conjecture Brylinsky-Kashiwara Elias-W. 2014,(multiplicity “ Py,xp1qq Beilinson-Bernstein following
Soergel 1990
Jantzen conjecture Beilinson-Bernstein W. 2016 following(graded multiplicity “ Py,xpvq) Soergel 2008,
Kübel 2012
Geometric proofs: D-modules, perverse sheaves, weights…
Algebraic proofs: “shadows of Hodge theory”,i.e. invariant forms (“geometric structures”)still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann
39
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potted history
Geometric proof (’80s) Algebraic proof (’10s)
Kazhdan-Lusztig conjecture Brylinsky-Kashiwara Elias-W. 2014,(multiplicity “ Py,xp1qq Beilinson-Bernstein following
Soergel 1990
Jantzen conjecture Beilinson-Bernstein W. 2016 following(graded multiplicity “ Py,xpvq) Soergel 2008,
Kübel 2012
Geometric proofs: D-modules, perverse sheaves, weights…
Algebraic proofs: “shadows of Hodge theory”,i.e. invariant forms (“geometric structures”)still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann
39
![Page 93: Representation theory and geometry · maschke’stheorem Maschke(1897) AnyrepresentationVofafinitegroupGoverR orC issemi-simple. Observation1:IfVhasapositive-definiteG-invariantgeometric](https://reader033.vdocuments.pub/reader033/viewer/2022050520/5fa3e56a0fbaf92cc155a96e/html5/thumbnails/93.jpg)
potted history
Geometric proof (’80s) Algebraic proof (’10s)
Kazhdan-Lusztig conjecture Brylinsky-Kashiwara Elias-W. 2014,(multiplicity “ Py,xp1qq Beilinson-Bernstein following
Soergel 1990
Jantzen conjecture Beilinson-Bernstein W. 2016 following(graded multiplicity “ Py,xpvq) Soergel 2008,
Kübel 2012
Geometric proofs: D-modules, perverse sheaves, weights…
Algebraic proofs: “shadows of Hodge theory”,i.e. invariant forms (“geometric structures”)still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann
39
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shadows of hodge theory
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beyond weyl groups
Weyl groups Ă Real reflection groups Ă Coxeter groups
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the coinvariant ring
Let W denote a real reflection group acting on hR.
Example
Symˆ ˙
ýR2 or Symˆ ˙
ýR3.
Let R denote the polynomial functions on hR. We view R as gradedwith h˚
R in degree 2.
Let RW` denote the W-invariants of positive degree. Set
H :“ R{pRW`q.
Remark
If W is the Weyl group of a complex semi-simple Lie algebra g, then His isomorphic to the cohomology of the flag variety of g (the “Borelisomorphism”).
42
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the coinvariant ring
Let W denote a real reflection group acting on hR.
Example
Symˆ ˙
ýR2 or Symˆ ˙
ýR3.
Let R denote the polynomial functions on hR. We view R as gradedwith h˚
R in degree 2.
Let RW` denote the W-invariants of positive degree. Set
H :“ R{pRW`q.
Remark
If W is the Weyl group of a complex semi-simple Lie algebra g, then His isomorphic to the cohomology of the flag variety of g (the “Borelisomorphism”).
42
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the coinvariant ring
Let W denote a real reflection group acting on hR.
Example
Symˆ ˙
ýR2 or Symˆ ˙
ýR3.
Let R denote the polynomial functions on hR. We view R as gradedwith h˚
R in degree 2.
Let RW` denote the W-invariants of positive degree. Set
H :“ R{pRW`q.
Remark
If W is the Weyl group of a complex semi-simple Lie algebra g, then His isomorphic to the cohomology of the flag variety of g (the “Borelisomorphism”).
42
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the coinvariant ring
Let W denote a real reflection group acting on hR.
Example
Symˆ ˙
ýR2 or Symˆ ˙
ýR3.
Let R denote the polynomial functions on hR. We view R as gradedwith h˚
R in degree 2.
Let RW` denote the W-invariants of positive degree. Set
H :“ R{pRW`q.
Remark
If W is the Weyl group of a complex semi-simple Lie algebra g, then His isomorphic to the cohomology of the flag variety of g (the “Borelisomorphism”).
42
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the coinvariant ring
Let W denote a real reflection group acting on hR.
Example
Symˆ ˙
ýR2 or Symˆ ˙
ýR3.
Let R denote the polynomial functions on hR. We view R as gradedwith h˚
R in degree 2.
Let RW` denote the W-invariants of positive degree. Set
H :“ R{pRW`q.
Remark
If W is the Weyl group of a complex semi-simple Lie algebra g, then His isomorphic to the cohomology of the flag variety of g (the “Borelisomorphism”).
42
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the coinvariant ring
Let W denote a real reflection group acting on hR.
Example
Symˆ ˙
ýR2 or Symˆ ˙
ýR3.
Let R denote the polynomial functions on hR. We view R as gradedwith h˚
R in degree 2.
Let RW` denote the W-invariants of positive degree. Set
H :“ R{pRW`q.
Remark
If W is the Weyl group of a complex semi-simple Lie algebra g, then His isomorphic to the cohomology of the flag variety of g (the “Borelisomorphism”).
42
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the invariant form
H :“ R{pRW`q
d :“ number of reflecting hyperplanes in hR
“complex dimension of flag variety”
There exists a unique (up to scalar) bilinear form
x´,´y : Hd´‚ ˆ Hd`‚ Ñ R
satisfying xγc, c1y “ xc, γc1y for all γ, c, c1 P H (the invariant form).
Remark
x´,´y is the analogue of the intersection form on cohomology.
43
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the invariant form
H :“ R{pRW`q
d :“ number of reflecting hyperplanes in hR
“complex dimension of flag variety”
There exists a unique (up to scalar) bilinear form
x´,´y : Hd´‚ ˆ Hd`‚ Ñ R
satisfying xγc, c1y “ xc, γc1y for all γ, c, c1 P H (the invariant form).
Remark
x´,´y is the analogue of the intersection form on cohomology.
43
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the invariant form
H :“ R{pRW`q
d :“ number of reflecting hyperplanes in hR
“complex dimension of flag variety”
There exists a unique (up to scalar) bilinear form
x´,´y : Hd´‚ ˆ Hd`‚ Ñ R
satisfying xγc, c1y “ xc, γc1y for all γ, c, c1 P H (the invariant form).
Remark
x´,´y is the analogue of the intersection form on cohomology.
43
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hodge theory for the coinvariant ring
There exists an open cone K Ă h˚R (“Kähler cone”).
Theorem (toy model)For all γ P K and i ě 0:
(a) (Hard Lefschetz) Multiplication by γi induces an isomor-phism
Hd´i „ÝÑ Hd`i
(b) (Hodge-Riemann bilinear relations) The form pc, c1q “
xc, γic1y on Hd´i is p´1q?-definite on the kernel of γi`1.
The theorem is identical to the hard Lefschetz theorem andHodge-Riemann bilinear relations in complex algebraic geometry.In the Weyl group case the theorem can be deduced from complexalgebraic geometry, but not in general.
H8
H6
H4
H2
H0
γ¨
γ¨
γ¨
γ¨
„„
44
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hodge theory for the coinvariant ring
There exists an open cone K Ă h˚R (“Kähler cone”).
Theorem (toy model)For all γ P K and i ě 0:
(a) (Hard Lefschetz) Multiplication by γi induces an isomor-phism
Hd´i „ÝÑ Hd`i
(b) (Hodge-Riemann bilinear relations) The form pc, c1q “
xc, γic1y on Hd´i is p´1q?-definite on the kernel of γi`1.
The theorem is identical to the hard Lefschetz theorem andHodge-Riemann bilinear relations in complex algebraic geometry.In the Weyl group case the theorem can be deduced from complexalgebraic geometry, but not in general.
H8
H6
H4
H2
H0
γ¨
γ¨
γ¨
γ¨
„„
44
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hodge theory for the coinvariant ring
There exists an open cone K Ă h˚R (“Kähler cone”).
Theorem (toy model)For all γ P K and i ě 0:
(a) (Hard Lefschetz) Multiplication by γi induces an isomor-phism
Hd´i „ÝÑ Hd`i
(b) (Hodge-Riemann bilinear relations) The form pc, c1q “
xc, γic1y on Hd´i is p´1q?-definite on the kernel of γi`1.
The theorem is identical to the hard Lefschetz theorem andHodge-Riemann bilinear relations in complex algebraic geometry.In the Weyl group case the theorem can be deduced from complexalgebraic geometry, but not in general.
H8
H6
H4
H2
H0
γ¨
γ¨
γ¨
γ¨
„„
44
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hodge theory for the coinvariant ring
There exists an open cone K Ă h˚R (“Kähler cone”).
Theorem (toy model)For all γ P K and i ě 0:
(a) (Hard Lefschetz) Multiplication by γi induces an isomor-phism
Hd´i „ÝÑ Hd`i
(b) (Hodge-Riemann bilinear relations) The form pc, c1q “
xc, γic1y on Hd´i is p´1q?-definite on the kernel of γi`1.
The theorem is identical to the hard Lefschetz theorem andHodge-Riemann bilinear relations in complex algebraic geometry.In the Weyl group case the theorem can be deduced from complexalgebraic geometry, but not in general.
H8
H6
H4
H2
H0
γ¨
γ¨
γ¨
γ¨
„„
44
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connection to the kazhdan-lusztig conjecture
In 1990 Soergel defined graded H-modules Hw for all w P W. Todaythey are known as “Soergel modules”. In the Weyl group case, heproved that the Kazhdan-Lusztig conjecture is equivalent to
dimR Hw “ÿ
yPWPy,wp1q. (1)
By appeal to the Decomposition Theorem (a deep theorem inalgebraic geometry) he deduced the equality. In doing so he wasable to identify the Hw with the intersection cohomology of Schubertvarieties.
We provided an algebraic proof of (1) as a consequence of
Theorem (Elias-W.)
The hard Lefschetz and Hodge-Riemann relations hold for Hw.
45
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connection to the kazhdan-lusztig conjecture
In 1990 Soergel defined graded H-modules Hw for all w P W. Todaythey are known as “Soergel modules”. In the Weyl group case, heproved that the Kazhdan-Lusztig conjecture is equivalent to
dimR Hw “ÿ
yPWPy,wp1q. (1)
By appeal to the Decomposition Theorem (a deep theorem inalgebraic geometry) he deduced the equality. In doing so he wasable to identify the Hw with the intersection cohomology of Schubertvarieties.
We provided an algebraic proof of (1) as a consequence of
Theorem (Elias-W.)
The hard Lefschetz and Hodge-Riemann relations hold for Hw.
45
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connection to the kazhdan-lusztig conjecture
In 1990 Soergel defined graded H-modules Hw for all w P W. Todaythey are known as “Soergel modules”. In the Weyl group case, heproved that the Kazhdan-Lusztig conjecture is equivalent to
dimR Hw “ÿ
yPWPy,wp1q. (1)
By appeal to the Decomposition Theorem (a deep theorem inalgebraic geometry) he deduced the equality. In doing so he wasable to identify the Hw with the intersection cohomology of Schubertvarieties.
We provided an algebraic proof of (1) as a consequence of
Theorem (Elias-W.)
The hard Lefschetz and Hodge-Riemann relations hold for Hw.45
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features of the proof
There is a resemblance to the semi-simple world:
(a) The invariant form x´,´y is unique up to scalar and satisfies theHodge-Riemann relations (“uniqueness of geometric structure”).
(b) The invariant form is our main tool in proving that Soergelmodules decompose as they should (“semi-simplicity viaintroduction of geometric structure”).
Ideas of de Cataldo and Migliorini provide several useful clues.
Diagrammatic algebra crucial to calculate and discover correctstatements.
46
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features of the proof
There is a resemblance to the semi-simple world:
(a) The invariant form x´,´y is unique up to scalar and satisfies theHodge-Riemann relations (“uniqueness of geometric structure”).
(b) The invariant form is our main tool in proving that Soergelmodules decompose as they should (“semi-simplicity viaintroduction of geometric structure”).
Ideas of de Cataldo and Migliorini provide several useful clues.
Diagrammatic algebra crucial to calculate and discover correctstatements.
46
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features of the proof
There is a resemblance to the semi-simple world:
(a) The invariant form x´,´y is unique up to scalar and satisfies theHodge-Riemann relations (“uniqueness of geometric structure”).
(b) The invariant form is our main tool in proving that Soergelmodules decompose as they should (“semi-simplicity viaintroduction of geometric structure”).
Ideas of de Cataldo and Migliorini provide several useful clues.
Diagrammatic algebra crucial to calculate and discover correctstatements.
46
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features of the proof
There is a resemblance to the semi-simple world:
(a) The invariant form x´,´y is unique up to scalar and satisfies theHodge-Riemann relations (“uniqueness of geometric structure”).
(b) The invariant form is our main tool in proving that Soergelmodules decompose as they should (“semi-simplicity viaintroduction of geometric structure”).
Ideas of de Cataldo and Migliorini provide several useful clues.
Diagrammatic algebra crucial to calculate and discover correctstatements.
46
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features of the proof
There is a resemblance to the semi-simple world:
(a) The invariant form x´,´y is unique up to scalar and satisfies theHodge-Riemann relations (“uniqueness of geometric structure”).
(b) The invariant form is our main tool in proving that Soergelmodules decompose as they should (“semi-simplicity viaintroduction of geometric structure”).
Ideas of de Cataldo and Migliorini provide several useful clues.
Diagrammatic algebra crucial to calculate and discover correctstatements.
46
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kazhdan-lusztig positivity
Kazhdan-Lusztig polynomials are defined for any pair of elements ina Coxeter group. The Kazhdan-Luszig positivity conjecture (1979) isthe statement that their coefficients are always non-negative.
Corollary (Elias-W. 2013)
The Kazhdan-Lusztig positivity conjecture holds.
A mystery for the 21st century?
Similar structures arise in the theory of non-rational polytopes (dueto McMullen, Braden-Lunts, Karu, …) and in recent work ofApridisato-Huh-Katz on matroids. Why?
47
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kazhdan-lusztig positivity
Kazhdan-Lusztig polynomials are defined for any pair of elements ina Coxeter group. The Kazhdan-Luszig positivity conjecture (1979) isthe statement that their coefficients are always non-negative.
Corollary (Elias-W. 2013)
The Kazhdan-Lusztig positivity conjecture holds.
A mystery for the 21st century?
Similar structures arise in the theory of non-rational polytopes (dueto McMullen, Braden-Lunts, Karu, …) and in recent work ofApridisato-Huh-Katz on matroids. Why?
47
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kazhdan-lusztig positivity
Kazhdan-Lusztig polynomials are defined for any pair of elements ina Coxeter group. The Kazhdan-Luszig positivity conjecture (1979) isthe statement that their coefficients are always non-negative.
Corollary (Elias-W. 2013)
The Kazhdan-Lusztig positivity conjecture holds.
A mystery for the 21st century?
Similar structures arise in the theory of non-rational polytopes (dueto McMullen, Braden-Lunts, Karu, …) and in recent work ofApridisato-Huh-Katz on matroids. Why?
47
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modular representations
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modular representations
There are analogies between infinite-dimensional representations ofLie algebras, and modular representations of algebraic groups. Theanalogue of the Kazhdan-Lusztig conjecture in this setting is theLusztig conjecture (1980).
The approach of Soergel is also fruitful for studying modularrepresentations. A major source of difficulty is that signature nolonger makes sense, and Kazhdan-Lusztig like formulas do notalways hold (such questions are tied to deciding when Lusztig’sconjecture holds).
Invariant forms (now defined over the integers) still play a decisiverole in the theory.
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modular representations
There are analogies between infinite-dimensional representations ofLie algebras, and modular representations of algebraic groups. Theanalogue of the Kazhdan-Lusztig conjecture in this setting is theLusztig conjecture (1980).
The approach of Soergel is also fruitful for studying modularrepresentations. A major source of difficulty is that signature nolonger makes sense, and Kazhdan-Lusztig like formulas do notalways hold (such questions are tied to deciding when Lusztig’sconjecture holds).
Invariant forms (now defined over the integers) still play a decisiverole in the theory.
49
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modular representations
There are analogies between infinite-dimensional representations ofLie algebras, and modular representations of algebraic groups. Theanalogue of the Kazhdan-Lusztig conjecture in this setting is theLusztig conjecture (1980).
The approach of Soergel is also fruitful for studying modularrepresentations. A major source of difficulty is that signature nolonger makes sense, and Kazhdan-Lusztig like formulas do notalways hold (such questions are tied to deciding when Lusztig’sconjecture holds).
Invariant forms (now defined over the integers) still play a decisiverole in the theory.
49
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lusztig conjecture
Lusztig conjecture (1980)
r∆̂As “ÿ
BqA,Bp1qrL̂Bs
(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraicgroups in characteristic p.
(b) Weyl group⇝ affine Weyl group.(c) True for large p depending on root system (Kashiwara-Tanisaki,
Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g.true for p ą 10100 for SL8.
(d) False for primes growing exponentially in the rank (W. 2014,following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.
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lusztig conjecture
Lusztig conjecture (1980)
r∆̂As “ÿ
BqA,Bp1qrL̂Bs
(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraicgroups in characteristic p.
(b) Weyl group⇝ affine Weyl group.(c) True for large p depending on root system (Kashiwara-Tanisaki,
Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g.true for p ą 10100 for SL8.
(d) False for primes growing exponentially in the rank (W. 2014,following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.
50
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lusztig conjecture
Lusztig conjecture (1980)
r∆̂As “ÿ
BqA,Bp1qrL̂Bs
(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraicgroups in characteristic p.
(b) Weyl group⇝ affine Weyl group.
(c) True for large p depending on root system (Kashiwara-Tanisaki,Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g.true for p ą 10100 for SL8.
(d) False for primes growing exponentially in the rank (W. 2014,following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.
50
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lusztig conjecture
Lusztig conjecture (1980)
r∆̂As “ÿ
BqA,Bp1qrL̂Bs
(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraicgroups in characteristic p.
(b) Weyl group⇝ affine Weyl group.(c) True for large p depending on root system (Kashiwara-Tanisaki,
Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g.true for p ą 10100 for SL8.
(d) False for primes growing exponentially in the rank (W. 2014,following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.
50
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lusztig conjecture
Lusztig conjecture (1980)
r∆̂As “ÿ
BqA,Bp1qrL̂Bs
(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraicgroups in characteristic p.
(b) Weyl group⇝ affine Weyl group.(c) True for large p depending on root system (Kashiwara-Tanisaki,
Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g.true for p ą 10100 for SL8.
(d) False for primes growing exponentially in the rank (W. 2014,following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.
50
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new character formula
Intersection cohomology sheaves⇝ parity sheaves (Soergel,Juteau-Mautner-W.).
Leads to p-Kazhdan-Lusztig polynomails pqA,B.
Riche-W. (2018)
r∆̂As “ÿ
B
pqA,Bp1qrL̂Bs
Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.pqA,B are computable via diagrammatic algebra + computer.
51
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new character formula
Intersection cohomology sheaves⇝ parity sheaves (Soergel,Juteau-Mautner-W.).
Leads to p-Kazhdan-Lusztig polynomails pqA,B.
Riche-W. (2018)
r∆̂As “ÿ
B
pqA,Bp1qrL̂Bs
Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.pqA,B are computable via diagrammatic algebra + computer.
51
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new character formula
Intersection cohomology sheaves⇝ parity sheaves (Soergel,Juteau-Mautner-W.).
Leads to p-Kazhdan-Lusztig polynomails pqA,B.
Riche-W. (2018)
r∆̂As “ÿ
B
pqA,Bp1qrL̂Bs
Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.pqA,B are computable via diagrammatic algebra + computer.
51
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new character formula
Intersection cohomology sheaves⇝ parity sheaves (Soergel,Juteau-Mautner-W.).
Leads to p-Kazhdan-Lusztig polynomails pqA,B.
Riche-W. (2018)
r∆̂As “ÿ
B
pqA,Bp1qrL̂Bs
Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.
pqA,B are computable via diagrammatic algebra + computer.
51
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new character formula
Intersection cohomology sheaves⇝ parity sheaves (Soergel,Juteau-Mautner-W.).
Leads to p-Kazhdan-Lusztig polynomails pqA,B.
Riche-W. (2018)
r∆̂As “ÿ
B
pqA,Bp1qrL̂Bs
Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.pqA,B are computable via diagrammatic algebra + computer.
51
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billiards conjecture
V a simple representation of Sn over Q.
⇝ reduce modulo p to get modular representation Fp bZ V.
Basic problem
Determine multiplicities of simple modules in Fp bZ V.
The answer is only known for partitions with one or two rows!
The following video illustrates the “billiards conjecture” (Lusztig-W.2017), which predicts many new cases of this decompositionbehaviour for partitions with three rows.
The conjecture predicts that these numbers are given by a “discretedynamical system”...
Billiards and tilting characters:https://www.youtube.com/watch?v=Ru0Zys1Vvq4
52
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billiards conjecture
V a simple representation of Sn over Q.
⇝ reduce modulo p to get modular representation Fp bZ V.
Basic problem
Determine multiplicities of simple modules in Fp bZ V.
The answer is only known for partitions with one or two rows!
The following video illustrates the “billiards conjecture” (Lusztig-W.2017), which predicts many new cases of this decompositionbehaviour for partitions with three rows.
The conjecture predicts that these numbers are given by a “discretedynamical system”...
Billiards and tilting characters:https://www.youtube.com/watch?v=Ru0Zys1Vvq4
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billiards conjecture
V a simple representation of Sn over Q.
⇝ reduce modulo p to get modular representation Fp bZ V.
Basic problem
Determine multiplicities of simple modules in Fp bZ V.
The answer is only known for partitions with one or two rows!
The following video illustrates the “billiards conjecture” (Lusztig-W.2017), which predicts many new cases of this decompositionbehaviour for partitions with three rows.
The conjecture predicts that these numbers are given by a “discretedynamical system”...
Billiards and tilting characters:https://www.youtube.com/watch?v=Ru0Zys1Vvq4
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![Page 137: Representation theory and geometry · maschke’stheorem Maschke(1897) AnyrepresentationVofafinitegroupGoverR orC issemi-simple. Observation1:IfVhasapositive-definiteG-invariantgeometric](https://reader033.vdocuments.pub/reader033/viewer/2022050520/5fa3e56a0fbaf92cc155a96e/html5/thumbnails/137.jpg)
billiards conjecture
V a simple representation of Sn over Q.
⇝ reduce modulo p to get modular representation Fp bZ V.
Basic problem
Determine multiplicities of simple modules in Fp bZ V.
The answer is only known for partitions with one or two rows!
The following video illustrates the “billiards conjecture” (Lusztig-W.2017), which predicts many new cases of this decompositionbehaviour for partitions with three rows.
The conjecture predicts that these numbers are given by a “discretedynamical system”...
Billiards and tilting characters:https://www.youtube.com/watch?v=Ru0Zys1Vvq4
52
![Page 138: Representation theory and geometry · maschke’stheorem Maschke(1897) AnyrepresentationVofafinitegroupGoverR orC issemi-simple. Observation1:IfVhasapositive-definiteG-invariantgeometric](https://reader033.vdocuments.pub/reader033/viewer/2022050520/5fa3e56a0fbaf92cc155a96e/html5/thumbnails/138.jpg)
billiards conjecture
V a simple representation of Sn over Q.
⇝ reduce modulo p to get modular representation Fp bZ V.
Basic problem
Determine multiplicities of simple modules in Fp bZ V.
The answer is only known for partitions with one or two rows!
The following video illustrates the “billiards conjecture” (Lusztig-W.2017), which predicts many new cases of this decompositionbehaviour for partitions with three rows.
The conjecture predicts that these numbers are given by a “discretedynamical system”...
Billiards and tilting characters:https://www.youtube.com/watch?v=Ru0Zys1Vvq4
52
![Page 139: Representation theory and geometry · maschke’stheorem Maschke(1897) AnyrepresentationVofafinitegroupGoverR orC issemi-simple. Observation1:IfVhasapositive-definiteG-invariantgeometric](https://reader033.vdocuments.pub/reader033/viewer/2022050520/5fa3e56a0fbaf92cc155a96e/html5/thumbnails/139.jpg)
billiards conjecture
V a simple representation of Sn over Q.
⇝ reduce modulo p to get modular representation Fp bZ V.
Basic problem
Determine multiplicities of simple modules in Fp bZ V.
The answer is only known for partitions with one or two rows!
The following video illustrates the “billiards conjecture” (Lusztig-W.2017), which predicts many new cases of this decompositionbehaviour for partitions with three rows.
The conjecture predicts that these numbers are given by a “discretedynamical system”...
Billiards and tilting characters:https://www.youtube.com/watch?v=Ru0Zys1Vvq4
52