Unipotent cells in Kac-Moody groups Bernard Leclerc joint with - PowerPoint PPT Presentation

The group N U ( n ) : universal enveloping algebra of n . It is generated by e i ( i ∈ I ), with relations 1 − c ij � ( − 1) k e ( k ) e j e (1 − c ij − k ) = 0 , ( i � = j ) . i i k =0 U ( n ) is a cocommutative Hopf algebra, with an R + -grading given by deg( e i ) := α i . gr := � Let U ( n ) ∗ d ∈ R + U ( n ) ∗ d be the graded dual. This is a commutative Hopf algebra. Let N := maxSpec( U ( n ) ∗ gr ) = Hom alg ( U ( n ) ∗ gr , C ). This is the pro-unipotent pro-group with Lie algebra � � n = g α . α ∈ ∆ +

The group N U ( n ) : universal enveloping algebra of n . It is generated by e i ( i ∈ I ), with relations 1 − c ij � ( − 1) k e ( k ) e j e (1 − c ij − k ) = 0 , ( i � = j ) . i i k =0 U ( n ) is a cocommutative Hopf algebra, with an R + -grading given by deg( e i ) := α i . gr := � Let U ( n ) ∗ d ∈ R + U ( n ) ∗ d be the graded dual. This is a commutative Hopf algebra. Let N := maxSpec( U ( n ) ∗ gr ) = Hom alg ( U ( n ) ∗ gr , C ). This is the pro-unipotent pro-group with Lie algebra � � n = g α . α ∈ ∆ + By construction, U ( n ) ∗ gr = C [ N ].

The unipotent subgroup N ( w )

The unipotent subgroup N ( w ) Let N ( w ) be the subgroup of N with Lie algebra n ( w ).

The unipotent subgroup N ( w ) Let N ( w ) be the subgroup of N with Lie algebra n ( w ). Let N ′ ( w ) be the subgroup of N with Lie algebra � n ′ ( w ) := g α ⊂ � n . α �∈ ∆ w

The unipotent subgroup N ( w ) Let N ( w ) be the subgroup of N with Lie algebra n ( w ). Let N ′ ( w ) be the subgroup of N with Lie algebra � n ′ ( w ) := g α ⊂ � n . α �∈ ∆ w Multiplication yields a bijection N ( w ) × N ′ ( w ) ∼ → N .

The unipotent subgroup N ( w ) Let N ( w ) be the subgroup of N with Lie algebra n ( w ). Let N ′ ( w ) be the subgroup of N with Lie algebra � n ′ ( w ) := g α ⊂ � n . α �∈ ∆ w Multiplication yields a bijection N ( w ) × N ′ ( w ) ∼ → N . Proposition The coordinate ring C [ N ( w )] is isomorphic to the invariant subring � � C [ N ] N ′ ( w ) = f ∈ C [ N ] | f ( nn ′ ) = f ( n ) , n ∈ N , n ′ ∈ N ( w )

The Kac-Moody group G

The Kac-Moody group G Let G be the group attached to g by Kac-Peterson.

The Kac-Moody group G Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system ( G , Norm G ( H ) , N + , N − , H ) , where Lie( H ) = h , Lie( N + ) = n + , and Lie( N − ) = n − .

The Kac-Moody group G Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system ( G , Norm G ( H ) , N + , N − , H ) , where Lie( H ) = h , Lie( N + ) = n + , and Lie( N − ) = n − . Note : In general N �⊂ G . They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N + = N ∩ G .

The Kac-Moody group G Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system ( G , Norm G ( H ) , N + , N − , H ) , where Lie( H ) = h , Lie( N + ) = n + , and Lie( N − ) = n − . Note : In general N �⊂ G . They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N + = N ∩ G . We have Norm G ( H ) / H ∼ = W .

The Kac-Moody group G Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system ( G , Norm G ( H ) , N + , N − , H ) , where Lie( H ) = h , Lie( N + ) = n + , and Lie( N − ) = n − . Note : In general N �⊂ G . They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N + = N ∩ G . We have Norm G ( H ) / H ∼ = W . For i ∈ I , put s i := exp( f i ) exp( − e i ) exp( f i ) ∈ Norm G ( H ) .

The Kac-Moody group G Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system ( G , Norm G ( H ) , N + , N − , H ) , where Lie( H ) = h , Lie( N + ) = n + , and Lie( N − ) = n − . Note : In general N �⊂ G . They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N + = N ∩ G . We have Norm G ( H ) / H ∼ = W . For i ∈ I , put s i := exp( f i ) exp( − e i ) exp( f i ) ∈ Norm G ( H ) . For w = s i r · · · s i 1 with ℓ ( w ) = r , put w = s i r · · · s i 1 , a representative of w in Norm G ( H ).

Examples

Examples If Q is of Dynkin type X n , then G = G X n ( C ) is a connected simply-connected algebraic group of type X n over C . (Ex: If X n = A n , G = SL ( n + 1 , C ).)

Examples If Q is of Dynkin type X n , then G = G X n ( C ) is a connected simply-connected algebraic group of type X n over C . (Ex: If X n = A n , G = SL ( n + 1 , C ).) If Q is of affine Dynkin type � X n , then G is a central extension by C ∗ of G X n ( C [ z , z − 1 ]).

Examples If Q is of Dynkin type X n , then G = G X n ( C ) is a connected simply-connected algebraic group of type X n over C . (Ex: If X n = A n , G = SL ( n + 1 , C ).) If Q is of affine Dynkin type � X n , then G is a central extension by C ∗ of G X n ( C [ z , z − 1 ]). Moreover, N + ≃ { g ∈ G X n ( C [ z ]) | g | z =0 ∈ N X n ( C ) } .

Examples If Q is of Dynkin type X n , then G = G X n ( C ) is a connected simply-connected algebraic group of type X n over C . (Ex: If X n = A n , G = SL ( n + 1 , C ).) If Q is of affine Dynkin type � X n , then G is a central extension by C ∗ of G X n ( C [ z , z − 1 ]). Moreover, N + ≃ { g ∈ G X n ( C [ z ]) | g | z =0 ∈ N X n ( C ) } . If Q is wild, no “concrete” realization of G is known.

Unipotent cells

Generalized minors

Generalized minors Let G 0 = N − HN + .

Generalized minors Let G 0 = N − HN + . Proposition G 0 is open dense in G. Every g ∈ G 0 has a unique factorization g = [ g ] − [ g ] 0 [ g ] + with [ g ] − ∈ N − , [ g ] 0 ∈ H , [ g ] + ∈ N + .

Generalized minors Let G 0 = N − HN + . Proposition G 0 is open dense in G. Every g ∈ G 0 has a unique factorization g = [ g ] − [ g ] 0 [ g ] + with [ g ] − ∈ N − , [ g ] 0 ∈ H , [ g ] + ∈ N + . For i ∈ I , let ̟ i ∈ h ∗ s. t. ̟ i ( h j ) = δ ij .

Generalized minors Let G 0 = N − HN + . Proposition G 0 is open dense in G. Every g ∈ G 0 has a unique factorization g = [ g ] − [ g ] 0 [ g ] + with [ g ] − ∈ N − , [ g ] 0 ∈ H , [ g ] + ∈ N + . For i ∈ I , let ̟ i ∈ h ∗ s. t. ̟ i ( h j ) = δ ij . Let x �→ x ̟ i denote the corresponding character of H .

Generalized minors Let G 0 = N − HN + . Proposition G 0 is open dense in G. Every g ∈ G 0 has a unique factorization g = [ g ] − [ g ] 0 [ g ] + with [ g ] − ∈ N − , [ g ] 0 ∈ H , [ g ] + ∈ N + . For i ∈ I , let ̟ i ∈ h ∗ s. t. ̟ i ( h j ) = δ ij . Let x �→ x ̟ i denote the corresponding character of H . There is a unique regular function ∆ ̟ i ,̟ i on G such that ∆ ̟ i ,̟ i ( g ) = [ g ] ̟ i ( g ∈ G 0 ) . 0

Generalized minors Let G 0 = N − HN + . Proposition G 0 is open dense in G. Every g ∈ G 0 has a unique factorization g = [ g ] − [ g ] 0 [ g ] + with [ g ] − ∈ N − , [ g ] 0 ∈ H , [ g ] + ∈ N + . For i ∈ I , let ̟ i ∈ h ∗ s. t. ̟ i ( h j ) = δ ij . Let x �→ x ̟ i denote the corresponding character of H . There is a unique regular function ∆ ̟ i ,̟ i on G such that ∆ ̟ i ,̟ i ( g ) = [ g ] ̟ i ( g ∈ G 0 ) . 0 For w ∈ W , set ∆ ̟ i , w ( ̟ i ) ( g ) := ∆ ̟ i ,̟ i ( gw ).

Generalized minors Let G 0 = N − HN + . Proposition G 0 is open dense in G. Every g ∈ G 0 has a unique factorization g = [ g ] − [ g ] 0 [ g ] + with [ g ] − ∈ N − , [ g ] 0 ∈ H , [ g ] + ∈ N + . For i ∈ I , let ̟ i ∈ h ∗ s. t. ̟ i ( h j ) = δ ij . Let x �→ x ̟ i denote the corresponding character of H . There is a unique regular function ∆ ̟ i ,̟ i on G such that ∆ ̟ i ,̟ i ( g ) = [ g ] ̟ i ( g ∈ G 0 ) . 0 For w ∈ W , set ∆ ̟ i , w ( ̟ i ) ( g ) := ∆ ̟ i ,̟ i ( gw ). G 0 = { g ∈ G | ∆ ̟ i ,̟ i ( g ) � = 0 , i ∈ I } .

The unipotent cell N w

The unipotent cell N w Let B − = N − H .

The unipotent cell N w Let B − = N − H . The group G has a Bruhat decomposition � G = B − wB − . w ∈ W

The unipotent cell N w Let B − = N − H . The group G has a Bruhat decomposition � G = B − wB − . w ∈ W � B − wB − . For w ∈ W , define N w := N +

The unipotent cell N w Let B − = N − H . The group G has a Bruhat decomposition � G = B − wB − . w ∈ W � B − wB − . For w ∈ W , define N w := N + Let O w := { n ∈ N ( w ) | ∆ ̟ i , w − 1 ( ̟ i ) ( n ) � = 0 , i ∈ I } .

The unipotent cell N w Let B − = N − H . The group G has a Bruhat decomposition � G = B − wB − . w ∈ W � B − wB − . For w ∈ W , define N w := N + Let O w := { n ∈ N ( w ) | ∆ ̟ i , w − 1 ( ̟ i ) ( n ) � = 0 , i ∈ I } . Proposition ∼ → N w . It follows that C [ N w ] is the We have an isomorphism O w localization of C [ N ( w )] ≃ C [ N ] N ′ ( w ) at � ∆ w := ∆ ̟ i , w − 1 ( ̟ i ) . i ∈ I

Concrete calculations

Concrete calculations Set x i ( t ) := exp( te i ) ( t ∈ C , i ∈ I ).

Concrete calculations Set x i ( t ) := exp( te i ) ( t ∈ C , i ∈ I ). Let w = s i 1 · · · s i r be a reduced decomposition.

Concrete calculations Set x i ( t ) := exp( te i ) ( t ∈ C , i ∈ I ). Let w = s i 1 · · · s i r be a reduced decomposition. The image of the map ( C ∗ ) r → N given by ( t 1 , . . . , t r ) �→ x i 1 ( t 1 ) · · · x i r ( t r ) is a dense subset of N w .

Concrete calculations Set x i ( t ) := exp( te i ) ( t ∈ C , i ∈ I ). Let w = s i 1 · · · s i r be a reduced decomposition. The image of the map ( C ∗ ) r → N given by ( t 1 , . . . , t r ) �→ x i 1 ( t 1 ) · · · x i r ( t r ) is a dense subset of N w . If f ∈ C [ N w ], then ( t 1 , . . . , t r ) �→ f ( x i 1 ( t 1 ) · · · x i r ( t r )) is a polynomial function, which completely determines f .

Concrete calculations Set x i ( t ) := exp( te i ) ( t ∈ C , i ∈ I ). Let w = s i 1 · · · s i r be a reduced decomposition. The image of the map ( C ∗ ) r → N given by ( t 1 , . . . , t r ) �→ x i 1 ( t 1 ) · · · x i r ( t r ) is a dense subset of N w . If f ∈ C [ N w ], then ( t 1 , . . . , t r ) �→ f ( x i 1 ( t 1 ) · · · x i r ( t r )) is a polynomial function, which completely determines f . Problem How to calculate f ( x i 1 ( t 1 ) · · · x i r ( t r )) , for example when f is a generalized minor ?

Preprojective algebras and semicanonical bases

The preprojective algebra

The preprojective algebra Λ : the preprojective algebra attached to Q .

The preprojective algebra Λ : the preprojective algebra attached to Q . nil(Λ) : category of finite-dimensional nilpotent Λ-modules.

The preprojective algebra Λ : the preprojective algebra attached to Q . nil(Λ) : category of finite-dimensional nilpotent Λ-modules. For M ∈ nil(Λ) and i := ( i 1 , . . . , i k ), let F M , i be the variety of type i composition series of M .

The preprojective algebra Λ : the preprojective algebra attached to Q . nil(Λ) : category of finite-dimensional nilpotent Λ-modules. For M ∈ nil(Λ) and i := ( i 1 , . . . , i k ), let F M , i be the variety of type i composition series of M . χ M , i := χ ( F M , i ) ∈ Z (Euler characteristic).

The preprojective algebra Λ : the preprojective algebra attached to Q . nil(Λ) : category of finite-dimensional nilpotent Λ-modules. For M ∈ nil(Λ) and i := ( i 1 , . . . , i k ), let F M , i be the variety of type i composition series of M . χ M , i := χ ( F M , i ) ∈ Z (Euler characteristic). Theorem (Lusztig, Geiss-L-Schr¨ oer) There exits a unique ϕ M ∈ C [ N ] such that for all j = ( j 1 , . . . , j k ) � χ M , j a t 1 a 1 · · · t ka k ϕ M ( x j 1 ( t 1 ) · · · x j k ( t k )) = a 1 ! · · · a k ! a ∈ N k where j a = ( j 1 , . . . , j 1 , . . . , j k , . . . , j k ) � �� � � �� � a 1 a k

An example

An example �� 2 Q = 1

� � � � An example �� 2 Q = 1 M 1 = 1 M 2 = 2 1 2 2 � � � � � � � � � � � � � � � 2 1 1 � � � � � 2

� � � � An example �� 2 Q = 1 M 1 = 1 M 2 = 2 1 2 2 � � � � � � � � � � � � � � � 2 1 1 � � � � � 2 ϕ M 1 ( x 2 ( t 1 ) x 1 ( t 2 ) x 2 ( t 3 ) x 1 ( t 4 )) = t 1 t 2 2 + 2 t 1 t 2 t 4 + t 1 t 2 4 + t 3 t 2 4

� � � � An example �� 2 Q = 1 M 1 = 1 M 2 = 2 1 2 2 � � � � � � � � � � � � � � � 2 1 1 � � � � � 2 ϕ M 1 ( x 2 ( t 1 ) x 1 ( t 2 ) x 2 ( t 3 ) x 1 ( t 4 )) = t 1 t 2 2 + 2 t 1 t 2 t 4 + t 1 t 2 4 + t 3 t 2 4 ϕ M 2 ( x 2 ( t 1 ) x 1 ( t 2 ) x 2 ( t 3 ) x 1 ( t 4 )) = t 1 t 2 2 t 3 3

Generalized minors

Generalized minors For M ∈ Mod Λ and i ∈ I , let soc i ( M ) be the S i -isotypic component of soc M .

Generalized minors For M ∈ Mod Λ and i ∈ I , let soc i ( M ) be the S i -isotypic component of soc M . For i = ( i 1 , . . . , i m ) there is a unique sequence 0 = M 0 ⊆ M 1 ⊆ · · · ⊆ M m ⊆ M such that soc i k ( M / M k − 1 ) ∼ = M k / M k − 1 for 1 ≤ k ≤ m . Define soc i ( M ) = M m .

Generalized minors For M ∈ Mod Λ and i ∈ I , let soc i ( M ) be the S i -isotypic component of soc M . For i = ( i 1 , . . . , i m ) there is a unique sequence 0 = M 0 ⊆ M 1 ⊆ · · · ⊆ M m ⊆ M such that soc i k ( M / M k − 1 ) ∼ = M k / M k − 1 for 1 ≤ k ≤ m . Define soc i ( M ) = M m . � I k : injective envelope of S k (infinite-dimensional).

Generalized minors For M ∈ Mod Λ and i ∈ I , let soc i ( M ) be the S i -isotypic component of soc M . For i = ( i 1 , . . . , i m ) there is a unique sequence 0 = M 0 ⊆ M 1 ⊆ · · · ⊆ M m ⊆ M such that soc i k ( M / M k − 1 ) ∼ = M k / M k − 1 for 1 ≤ k ≤ m . Define soc i ( M ) = M m . � I k : injective envelope of S k (infinite-dimensional). Proposition (Geiss-L-Schr¨ oer) Let i be a reduced word for w − 1 ∈ W . Then ∆ ̟ k , w ( ̟ k ) = ϕ soc i ( b I k ) .