Formal group laws and cohomology of quiver varieties

Transcription

1 Formal group laws and cohomology of quiver varieties Gufang Zhao IMJ-PRG Séminaire Caen Cergy Clermont Paris - Théorie des Représentations Oct. 31, 2014, Paris Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 0 / 24

2 Outline 1 Motivation Affine Hecke algebra Quantum groups Generalizations 2 Warm-up: formal affine Hecke algebras Formal group laws Formal affine Hecke algebra Geometric study of the formal affine Hecke algebras 3 Elliptic affine Hecke algebras Residue construction Equivariant elliptic cohomology Geometric study of the elliptic affine Hecke algebra 4 Hall algebras of a quiver Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 0 / 24

3 Motivation 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 : e λ eλ e s α(λ) e α 1 q eλ e s α(λ)+α e α. 1 The Hecke algebra: H aff End Z[q ± ](Z[q ± ][Λ]) is generated by Z[q ± ][Λ] and T α, α. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 1 / 24

4 Motivation Affine Hecke algebra Affine Hecke algebra: geometric construction B := G/B. K Gm G(T B) Z[q ± ][Λ]; π : Ñ := T B N the Springer resolution; for each x N, let B x := π 1 (x) be the Springer fiber; Z := Ñ N Ñ the Steinberg variety, with projections to the ith factor p i : Z T B, i = 1, 2; Theorem (Demazure 1974, Lusztig 1990, etc) 1 For each simple root α, there is a class T α K G Gm (Z) such that the operator p 1 (T α p 2 ) : K G m G(T B) K Gm G(T B) acts by T α. 2 This induces an isomorphism K Gm G(Z) H aff. 3 For each x N, K(B x ) admits an action by H aff. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 2 / 24

5 Motivation Quantum groups Quantum groups Let Q = (I, H) be a quiver without edge-loop, and let g be the Lie algebra. The Yangian Y (Lg) and the quantum loop algebra U q (Lg), deforming respectively the enveloping algebras of g[u] and g[u ± ]. For any v and w N I, let M(v, w) be the Nakajima quiver variety. Theorem (Nakajima 1999, Varagnolo 2000) There are representations 1 Y (Lg) End( v N IH G w G m (M(v, w))); 2 U q (Lg) End( v N IK Gw G m (M(v, w))). These are highest weight integrable representations. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 3 / 24

6 Motivation Generalizations Oriented cohomology theories Definition (Levine-Morel 2007) An oriented cohomology theory (OTC) on Sm k is given by the following. D1 A functor A from Sm op to the cat. of comm. rings with unit. k D2 For any projective morphism f : Y X in Sm k, a homomorphism of graded A (X)-modules f : A (Y) A (X), which is funtorial w.r.t. projective morphisms. These data satisfy: base change for transverse morphisms; a projective bundle formula; and extended homotopy property. For L X a line bundle, the first Chern class is defined as c A 1 (L) := s s (1 X ), where s : X L is the zero section. For any linear algebraic group G, using Borel construction one extend A to A G. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 4 / 24

7 Motivation Generalizations Motivating questions Let A be an OCT. Whether there is a generalized affine Hecke algebra acting on the A (B x ) by convolutions with cohomology classes on the Steinberg variety. Whether there is an algebra U A (generalized quantum group, depending on the oriented cohomology theory A ) that acts on v A G w G m 2(M(v, w))? What about the equivariant elliptic cohomology? Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 5 / 24

8 Table of Contents Warm-up: formal affine Hecke algebras 1 Motivation Affine Hecke algebra Quantum groups Generalizations 2 Warm-up: formal affine Hecke algebras Formal group laws Formal affine Hecke algebra Geometric study of the formal affine Hecke algebras 3 Elliptic affine Hecke algebras Residue construction Equivariant elliptic cohomology Geometric study of the elliptic affine Hecke algebra 4 Hall algebras of a quiver Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 5 / 24

9 Warm-up: formal affine Hecke algebras Formal group laws Formal group laws Let R be a commutative ring. A formal group law (FGL) on R is F(u, v) R[[u, v]] such that F(x, y) = F(y, x), F(x, 0) = 0, F(x, F(y, z)) = F(F(x, y), z). Let A be an OCT, let R := A (pt), then there is a formal group law F(u, v) R[[u, v]] such that for any X and line bundle L, M on X, we have c 1 (L M) = F(c 1 (L), c 1 (M)) A (X). Example: the K-theory has c K l (L) = 1 L 1, and formal group law is F(x, y) = x + y xy (multiplicative). Example, let E be an elliptic curve, with local coordinate l (a local analytic function around 0), then F(l(u), l(v)) = l(u + v) defines a FGL. The OCT with this FGL is called an elliptic cohomology. Let F x R[[x]] be the power series such that F(x, F x) = 0, and write x + F y = F(x, y) and x F y = x + F ( F y). Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 6 / 24

10 Warm-up: formal affine Hecke algebras Formal group laws Caution: Content May Be Explicit!!! Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 6 / 24

11 Warm-up: formal affine Hecke algebras Formal affine Hecke algebra Formal affine Hecke algebras Λ = rank-n free abelian group. Define the formal group algebra R[[Λ]] F := λ Λ R[[x λ ]]/ ( x 0, x λ + F x µ x λ+µ ). Γ = rank-1 free abelian group generated by γ Γ. Consider S := R[[Γ Λ]] F R[[Γ]] F [[Λ]] F. The action of the Weyl group on Λ induces an action of W on S. More precisely, w(x γ ) = x γ and w(x λ ) = x w(λ) for any λ Λ. Let Q F = S[ 1 x α ] α Φ + and Q F W := QF #W. For each w W, write δ w Q F W. Definition (Hoffnung Malagón-López Savage Zainoulline 2012) For simple root α, we define X F α := 1 x α 1 x α δ sα and T F α = x γ X α + δ α Q F W. The formal affine Demazure algebra D F Q F is the subalgebra of W generated by S and X α, α. The formal affine Hecke algebra H F Q F W generated by S and T α, α. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 7 / 24

12 Warm-up: formal affine Hecke algebras Formal affine Hecke algebra Formal affine Hecke algebras: properties For each w W, fix a reduced expression w = s i1 s il Write I w = (i 1,..., i l ) and write T Iw = T i1 T il. We have H F = {z w W u w T Iw u w Q F and z S S}. The action H F End R[[xγ ]](S) is well-defined and faithful. The operators T i and T j on S satisfy the following relations 1 T i T j = T j T i, if (s i s j ) 2 ( = 1 for s i and s j W; 2 T i T j T i T j T i T j = xγ 2 κji T j κ ij T i + κ j κ i ) x i+j, if (si s j ) 3 = 1 for s i and s j W. 3 T i T j T i T j T i T j = l(v) m } {{ }} {{ } ij 2 uvt ij Iv with the coefficients u ij v m ij times m ij times belonging to S for m ij = 2, 3, 4, 6. These operators satisfy the braid relation if and only if the formal group law is additive or multiplicative. (Bressler-Evens 1990.) There is a filtration of H F by R-submodule, inducing an isomorphism of graded rings Gr H F R Z H deg. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 8 / 24

13 Warm-up: formal affine Hecke algebras Formal affine Hecke algebra The formal affine Demazure algebra [Calméz-Zainoulline-Zhong 2012] We have D F = {z Q F z S S}. W For each w W, fix a reduced expression w = s i1 s il Write I w = (i 1,..., i l ) and write X Iw = X i1 X il. {X Iw } w W form a S-basis of D F. [Calméz-Zainoulline-Zhong 2014] For any simple root α, let P α be the corresponding parabolic group. Let p α : B G/P α be the natural projection. Identify A G (B) A T (pt) with S, we have p α p α = X α End(S). Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 9 / 24

14 Warm-up: formal affine Hecke algebras Geometric study of the formal affine Hecke algebras Formal affine Hecke algebras: convolution construction For simple root α, let J F α = T α µ α x γ κ α H F, κ α = 1 x α + 1 x α, µ α = x α x α. Let π : Z α Y α B B be the bundle projection, and π i : Y α B the ith projection, i = 1, 2. Let J α be π Ω 1 π 2 on Z α. Define J A α := c 1(J α ) c 1 (k q ) c 1 (J α k q ) A G G m (Z α ). Theorem (Z.-Zhong, 2014) 1 There is an R-algebra homomorphism Ψ A : H F End R (A G G m (Ñ)). 2 J F α H F is sent by Ψ A to convolution with the cohomology class J A α A G G m (Z α ). 3 For any x N, A (B x ) admits a natural action by H F. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 10 / 24

15 Warm-up: formal affine Hecke algebras Geometric study of the formal affine Hecke algebras Bivariant Riemann-Roch Let R Q, then l(t) R[[t]] such that F(l(x), l(y)) = l(x + y). Proposition (Panin-Smirnov) Let A be an OCT with FGL (R, F). There is an isomorphism of OCT s A (X) Ch (X) Z R sending c A (L) to l(cch(l)) for any line bundle L. 1 1 Let Z M M be a correspondence. RR : A (Z) Ch (Z) Z R, such that c A (L) l(cch 1 1 (L)) Td l(tm 2 ). There is an algebra isomorphism φ F,Fa : H F H Fa, T F l(x γ) δ i i such that l(x i ) + l(x i) l(x γ ) l(x i ) H F φ F,Fa H a F Ψ A A G G m (Z; Q) RR Ψ Ch Ch G G m (Z; Q) R Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 11 / 24

16 Table of Contents 1 Motivation Affine Hecke algebra Quantum groups Generalizations Elliptic affine Hecke algebras 2 Warm-up: formal affine Hecke algebras Formal group laws Formal affine Hecke algebra Geometric study of the formal affine Hecke algebras 3 Elliptic affine Hecke algebras Residue construction Equivariant elliptic cohomology Geometric study of the elliptic affine Hecke algebra 4 Hall algebras of a quiver Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 11 / 24

17 Elliptic affine Hecke algebras Caution: Work in progress. Handle with care!! Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 11 / 24

18 Elliptic affine Hecke algebras Residue construction 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,γ the divisor on A E E Z (Γ Λ ) defined by x α = x γ. For any open set U A/W E, consider rational sections of (S O E ) U #W, written as w W f w δ w where f w are rational sections of S regular ways from D α, such that R1 for any root α, each f w has a pole of order 1 along the divisor D α ; R2 α, the residues of f w and f sα w along D α differ by a minors sign; R3 for any α Σ(w), the local section f w vanishes along the divisor D α,γ. Theorem (Ginzburg-Kapranov-Vasserot 1998) The above conditions define a sheaf of associative algebra H on A E, called the elliptic affine Hecke algebra. The Demazure-Lusztig operator T α = sn(x γ) sn(x α ) + (1 sn(x γ) sn(x α ) )δ α, α. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 12 / 24

19 Elliptic affine Hecke algebras Equivariant elliptic cohomology Equivariant elliptic cohomology Come back to the category of topological spaces. Let E S be an arbitrary elliptic curve, together with a local coordinate l which extends to a rational section of a line bundle Π. The section l defines the structure of a cohomology theory A, 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 (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 13 / 24

20 Elliptic affine Hecke algebras 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 Ell 0 G (X) on A G. Ell 0 T (pt) O A T ; and Ell 0 G (pt) O A G. a A T, let T(a) = a AT A T T Ell 0 T (X) a Ell 0 T/T(a) (X T(a) ) 0 ; and Ell 0 T (X) a Q H T (X T(a), Q) 0. Denote Spec(Ell 0 G (X)) by AX. For any equivariant rank-n vector G bundle ξ : V X, there is a classifying map c ξ : A X G A GL n E (n). Define Θ(ξ) = cξ 1 (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 Ell 0 G (X) Θ(N XY) Ell 0 (Y, X). G For proper f : X Y, there is a f : Θ(Tf := TX f TY) Ell 0 G (Y). Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 14 / 24

21 Elliptic affine Hecke algebras 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 c ξ : A X G E(r) be the GKV-classifying map. Define the i-th l-chern class of V to be cξ 1 (σ i (l)) as a rational section of Ell (X). For any rank-r equivariant vector bundle ξ : V X, define cr GKV (V) := cξ 1 (ϑ r ) Θ(V) cξ 1 (Θ r ). It follows from the Thom isomorphism theorem that multiplication by (V) is equivalent to za z A : Θ(V) Ell(X). c GKV r Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 15 / 24

22 Elliptic affine Hecke algebras 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 ). Let B G/T; then A B G A T E n as scheme over A G = E n /W. 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 pt A is well-defined. Pα Proposition We have pα p α = X α = 1 ϑ(x α ) δ α ϑ(x α End( S). ) Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 16 / 24

23 Elliptic affine Hecke algebras 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, M 1 π T 1 M 1 p 1 Z p 2 T M 2 M 2. i 2 Define Ξ Z := Θ(p 2 ) Θ(p 1 π 1 T M 1 ) Θ(p 2 π 2 T M 2 ) 1 on A Z G. There is an action Ξ Z H om AG (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 is a representation. G Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 17 / 24

24 Elliptic affine Hecke algebras Geometric study of the elliptic affine Hecke algebra The elliptic affine Hecke algebras Theorem (Z.-Zhong, to appear) For each simple root α, the element J α := cgkv (J 1 α ) c GKV (J 1 α kq ) 1 cl 1 (k q) c l 1 (J α) as a rational section of Ξ(Z α ) acts by convolution as (1 l(x γ) l(x α ) )(s α + 1). As a sheaf on A G, H is locally free of rank = W 2. 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(x γ) sn(x α ) + (1 sn(x γ) sn(x α ) )δ α as in GKV 95. Away from the non-identity zero s of the sn(x α ), these operators and S generate H. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 18 / 24

25 Elliptic affine Hecke algebras Geometric study of the elliptic affine Hecke algebra Representations of the elliptic affine Hecke algebra Definition A representation of H is a coherent sheaf on A G, endowed with an action of H. Corollary Assume E is an elliptic curve over C. For any t E non-torsion, the simple H-representations supported on A G {t} are parameterized by triples (a, x, χ) where a A G, x N T(a,t) with T(a, t) T S 1 as before, and χ an irrep. of C(a, t, x), up to G-conjugation. (Here C(a, t, x) = G(a, t, x)/g 0 (a, t, x) is the component group, with G(a, t, x) G being the simultaneous centralizer of (a, t) and x.) Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 19 / 24

26 Table of Contents 1 Motivation Affine Hecke algebra Quantum groups Generalizations Hall algebras of a quiver 2 Warm-up: formal affine Hecke algebras Formal group laws Formal affine Hecke algebra Geometric study of the formal affine Hecke algebras 3 Elliptic affine Hecke algebras Residue construction Equivariant elliptic cohomology Geometric study of the elliptic affine Hecke algebra 4 Hall algebras of a quiver Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 19 / 24

27 Hall algebras of a quiver Cohomology of quiver varieties (R, F) any formal group law; Q = (I, H) any quiver. For each v N I, let Rep(Q, v) be the representation space. Let T = G m 2, acting on T Rep(Q, v); the 1st G m -factor of T scales Rep(Q, v), the 2nd factor scales the fibers of the cotangent bundle. As R[[t 1, t 2 ]]-module, A G v T (T Rep(Q, v)) R[[t 1, t 2 ]][[λ i s]] S v i I,s=1,...,v i, denoted by SH v. There is a Hall multiplication m S v 1,v 2 : SH v1 R[[t1,t 2 ]] SH v2 SH v1 +v 2 making it into an associative algebra. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 20 / 24

28 Hall algebras of a quiver The formal shuffle algebras The formal shuffle algebra is the N I -graded R[[t 1, t 2 ]]-algebra SH = v N ISH v with the multiplication as above. For any v 1 and v 2 N I. The multiplication SH v1 R[[t1,t 2 ]] SH v2 SH v1 +v 2 has formula f 1 (λ ) f 2 (λ ) R[[t 1, t 2 ]][[λ i s ]] S v 1 R[[t i I,s=1,...,v i 1, t 2 ]][[λ i t ]] S v 2 i I,t=1,...,v i 1 2 σ Sh(v 1,v 2 ) σ(f 1 f 2 fac 1 fac 2 ) R[[t 1, t 2 ]][[λ i j ]]S v 1 +v 2 i I,j=1,...,(v 1 +v 2 ) i. where fac 1 = v i v i 1 2 λ i t F λ i s+ F t 1 + F t 2 i I s=1 t=1 λ i t F λ i s and fac 2 = v i v j 1 2 i,j I s=1 t=1 (λ j t F λ i s + F t 1 ) a ij(λ j t F λ i s + F t 2 ) a ij. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 21 / 24

29 Hall algebras of a quiver Formal cohomological Hall algebras Let µ v : T Rep(Q, v) gl v be the moment map. Let P v = A T G v (µ 1 v (0)), and P = v N IP v. Theorem (Vasserot-Schiffmann 2008, Yang-Z., 2014) m P v 1,v 2 : P v1 P v2 P v1 +v 2 making P an associative algebra, such that. 1 There is an algebra homomorphism Θ : P SH; 2 There is an algebra homomorphism Φ : P End AT (pt)( v N IA T Gw (M(v, w))); 3 For simple roots α k of Q, let e k the corresponding dimension vector. Lagrangian C + k (v 1, w) M(v 1, w) M(v 1 + e k, w) and bundle L k. Let ξ αk be the natural representation of G ek. Then c 1 (ξ αk ) A T Gα (µ 1 α (0)) acts on v (A GLw T(M(v, w))) by convolution with c 1 (L k ). Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 22 / 24

30 Hall algebras of a quiver Speculations Let SP P be the subalgebra generated by A T Gek (µ 1 e k (0)) for simple roots α k. It is reasonable to expect that the algebra homomorphism Θ is an isomorphism onto its image when restricted to the subalgebra SP, and that the representation Φ factors through Θ. When Q is a quiver without edge loop, it is reasonable to expect that there is a "Drinfeld double" of SP, of which Nakajima quiver varieties provide highest weight intergrable representations. Gufang Zhao (IMJ-PRG) Formal group laws and quiver varieties Oct. 31, 2014, Paris 23 / 24

31 Thank You!!!