Elliptic affine Hecke algebra and representations A topological - PowerPoint PPT Presentation

Elliptic affine Hecke algebra and representations A topological method in geometric representation theory Joint with Changlong Zhong Gufang Zhao Institut de Mathématiques de Jussieu – Paris Rive Gauche Conference on Geometric Methods in Representation Theory Nov. 23, 2014, Iowa City Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 0 / 13

Outline Definition of the elliptic affine Hecke algebra 1 Equivariant elliptic cohomology 2 Geometric study of the elliptic affine Hecke algebra 3 Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 0 / 13

Definition of the elliptic affine Hecke algebra Affine Hecke algebra Let G be a simply connected complex algebraic group; T < B < G a maximal torus, and a Borel; Λ = the weight lattice; W = the Weyl group. W acts on Z [ q ± ][Λ] in the natural way; For λ ∈ Λ , write e λ ∈ Z [ q ± ][Λ] . For each simple root α , define the operator T α ∈ End Z [ q ± ] ( Z [ q ± ][Λ]) T α = qe α − q − 1 s α − q − q − 1 e α − 1 . e α − 1 The affine Hecke algebra: H aff ⊆ End Z [ q ± ] ( Z [ q ± ][Λ]) is generated by Z [ q ± ][Λ] and T α , α ∈ ∆ . Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 1 / 13

Definition of the elliptic affine Hecke algebra Affine Hecke algebra: geometric construction B := G / B . K G m × G ( T ∗ B ) � Z [ q ± ][Λ] ; π : � N := T ∗ B → N the Springer resolution; for each x ∈ N , let B x := π − 1 ( x ) be the Springer fiber; Z := � N × N � N the Steinberg variety, with projections to the i th factor p i : Z → T ∗ B , i = 1 , 2; Theorem (Demazure 1974, Lusztig 1990, etc) For each simple root α , there is a class T α ∈ K G × G m ( Z ) such that the 1 operator p 1 ∗ ( T α · p ∗ 2 − ) : K G m × G ( T ∗ B ) → K G m × G ( T ∗ B ) acts by T α . This induces an isomorphism K G m × G ( Z ) � H aff . 2 For each x ∈ N , K ( B x ) admits an action by H aff . 3 Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 2 / 13

Definition of the elliptic affine Hecke algebra Elliptic affine Hecke algebra Let E be an elliptic curve. Let A := E ⊗ Z Λ ∨ . Each root α gives a divisor D α of A . Let π : A → A / W and let S := π ∗ O A on A / W . Note that W acts on S . D α, q the divisor on A × E � E ⊗ Z ( Z ⊕ Λ ∨ ) defined by x α = q . For any open set U ⊆ A / W × E , consider rational sections of ( S ⊠ O E ) | U # W , written as � w ∈ W f w w where f w are rational sections of S regular ways from D α , such that R 1 for any root α , each f w has a pole of order ≤ 1 along the divisor D α ; R 2 ∀ α , the residues of f w and f s α w along D α differ by a minors sign; R 3 for any α ∈ w Φ + ∩ Φ − , the section f w vanishes along the divisor D α, q . Theorem (Ginzburg-Kapranov-Vasserot 1997) The above conditions define a sheaf of associative algebra H on A × E, called the elliptic affine Hecke algebra. H naturally acts on S ⊠ O E . The Demazure-Lusztig operator T α = sn ( q ) sn ( x α ) + ( 1 − sn ( q ) sn ( x α ) ) s α , α ∈ ∆ . Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 3 / 13

Definition of the elliptic affine Hecke algebra Motivating questions Is there a construction of the elliptic affine Hecke algebra as a convolution algebra? What are the irreducible representations of the elliptic affine Hecke algebra? Where does this algebra fit in the family of all these Hecke algebras? (Why should we care about it?) Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 4 / 13

Equivariant elliptic cohomology Table of Contents Definition of the elliptic affine Hecke algebra 1 Equivariant elliptic cohomology 2 Geometric study of the elliptic affine Hecke algebra 3 Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 4 / 13

Equivariant elliptic cohomology Equivariant elliptic cohomology Let E → S be an arbitrary elliptic curve, together with a local coordinate l (a rational section of a line bundle which vanishes of order 1 at the identity section). The local coordinate l defines the structure of a cohomology theory E ll ∗ , whose formal group law F ( u , v ) satisfies F ( l ( u ) , l ( v )) = l ( u + v ) . Let G be a compact Lie group with maximal torus T , let A G be the moduli space of semistable topologically trivial G -bundles on E ∨ . A T = E n where n = rank T , and A G = A T / W . Example When G = U n , we have A G � E ( n ) . Let Θ n be the big diagonal divisor (considered as a line bundle). It has a natural section denoted by ϑ n . When n = 1, Θ n is the divisor O ( − 0 ) and ϑ n is the Jacobi theta-function. Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 5 / 13

Equivariant elliptic cohomology Equivariant elliptic cohomology Theorem (Ginzburg-Kapranov-Vasserot 1995, Lurie 2005, etc) For any G-space X, there is a sheaf of comm. algebras E ll 0 G ( X ) on A G . E ll 0 T ( pt ) � O A T ; and E ll 0 G ( pt ) � O A G . ∀ a ∈ A T , let T ( a ) = ∩ a ∈ A T ′ ⊆ A T T ′ ⊆ T E ll 0 T ( X ) a � E ll 0 T / T ( a ) ( X T ( a ) ) 0 ; T ( X T ( a ) , Q ) ∧ There is a Chern character ch : E ll 0 T ( X ) ∧ a ⊗ Q � H ∗ 0 . Denote Spec ( E ll 0 G ( X )) by A X G . For any equivariant rank-n vector G → A GL n � E ( n ) . bundle ξ : V → X, there is a classifying map c ξ : A X Define Θ( ξ ) = c − 1 ξ (Θ n ) . Then this extends to Θ : K G ( X ) → Pic ( A X G ) . For any equivariant regular embedding X → Y, there is a Thom isomorphism E ll 0 G ( X ) ⊗ Θ( N X Y ) � E ll 0 G ( Y , X ) . For proper f : X → Y , there is a f ∗ : Θ( Tf := TX − f ∗ TY ) → E ll 0 G ( Y ) . Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 6 / 13

Equivariant elliptic cohomology Elliptic cohomology: Chern classes Recall that l is a local coordinate of E . For any r , i ∈ Z ≥ 0 , let σ i be the i -th elementary symmetric function. Then σ i ( l )( x 1 , . . . , x r ) := σ i ( l ( x 1 ) , . . . , l ( x r )) is a well-defined rational section of a line bundle on E ( r ) . For any G -equivariant rank- r vector bundle ξ : V → X , let G → E ( r ) be the GKV-classifying map. c ξ : A X Define the i -th l -Chern class of V to be c − 1 ξ ( σ i ( l )) as a rational section of E ll ∗ ( X ) . For any rank- r equivariant vector bundle ξ : V → X , define c GKV ( V ) := c − 1 ξ ( ϑ r ) ∈ Θ( V ) � c − 1 ξ (Θ r ) . r It follows from the Thom isomorphism theorem that multiplication by A ◦ z A ∗ : Θ( V ) ∨ → E ll ( X ) . c GKV ( V ) is equivalent to z ∗ r Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 7 / 13

Geometric study of the elliptic affine Hecke algebra Table of Contents Definition of the elliptic affine Hecke algebra 1 Equivariant elliptic cohomology 2 Geometric study of the elliptic affine Hecke algebra 3 Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 7 / 13

Geometric study of the elliptic affine Hecke algebra The elliptic affine Demazure algebras Let T be a torus. Recall Θ : Z [Λ] � K T ( pt ) → Pic ( A T ) . Example Let χ : T → S 1 be a character, then Θ( χ ) is the divisor A ker χ ⊆ A T . The pull-back section is denoted by ϑ χ . Definition Let � S := ⊕ λ ∈ Z [Λ] Θ( λ ) , with ring structure from the addition in Pic ( A T ) . G � A T � E n as scheme over A G = E n / W . Let B � G / T ; then A B For any simple root α , let p α : B → G / P α be the natural projection. The operator p α ∗ : Θ( Tp α ) � L α → O A pt P α is well-defined. Proposition 1 ϑ ( x α ) ∈ End ( � s α We have p ∗ α ◦ p α ∗ = X α = S ) . ϑ ( x α ) − Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 8 / 13

� � � Geometric study of the elliptic affine Hecke algebra Elliptic cohomology: Convolution For any smooth M 1 and M 2 , and a Lagrangian Z ⊆ T ∗ M 1 × T ∗ M 2 , p 2 � T ∗ M 2 T ∗ M 1 M 1 Z M 2 . π 1 p 1 i 2 2 T ∗ M 2 ) − 1 on A Z Define Ξ Z := Θ( p 2 ) ⊗ Θ( p ∗ 1 π ∗ 1 T ∗ M 1 ) ⊗ Θ( p ∗ 2 π ∗ G . There is an action Ξ Z → H om A G (Θ( T ∗ M 1 ) − 1 , Θ( T ∗ M 2 ) − 1 ) . When M 1 = M 2 and Z ◦ Z = Z , then Ξ Z is a sheaf of associative algebras and Θ( T ∗ M 2 ) − 1 on A M 2 G is a representation. Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 9 / 13

Geometric study of the elliptic affine Hecke algebra The elliptic affine Hecke algebras π : � N := T ∗ B → N the Springer resolution; Z := � N × N � N the Steinberg variety. Theorem (Z.-Zhong, to appear) There is an isomorphism Ξ( Z ) � H . For each simple root α , the element   c GKV c l ( J α ) 1 ( k q )       1 J α := ·   1 −   c l c GKV ( J α ⊗ k ∨ q ) 1 ( J α ) 1 as a rational section of Ξ( Z α ) acts by convolution as ( 1 − l ( q ) l ( x α ) )( s α + 1 ) . In particular, when E is an elliptic curve over C , Π = Ω E , and the local coordinate l = sn, we have J α − id acting by the same Demazure-Lusztig operator sn ( q ) sn ( x α ) + ( 1 − sn ( q ) sn ( x α ) ) s α as in GKV 95’. Gufang Zhao (Jussieu) Elliptic affine Hecke algebra Nov. 23, 2014, Iowa City 10 / 13