Toshiaki Shoji (Nagoya University) Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 1 / 1

Transcription

1 Character sheaves on a symmetric space and Kostka polynomials Toshiaki Shoji Nagoya University July 27, 2012, Osaka Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 1 / 1

2 Kostka polynomials K λ,µ (t) λ = (λ 1,..., λ k ) : partition of n λ i Z 0, λ 1 λ k 0, P n = {partitions of n} i λ i = n s λ (x) = s λ (x 1,..., x k ) Z[x 1,..., x k ] : Schur function P λ (x; t) = P λ (x 1,..., x k ; t) Z[x 1,..., x k ; t] : Hall-Littlewood function For λ, µ P n, K λ,µ (t): Kostka polynomial defined by s λ (x) = K λ,µ (t)p µ (x; t) µ P n K λ,µ (t) Z[t], (K λ,µ (t)) λ,µ Pn : transition matrix of two basis {s λ (x)}, {P µ (x; t)} of the space of homog. symm. poly. of degree n Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 2 / 1

3 Geometric realization of Kostka polynomials In 1981, Lusztig gave a geometric realization of Kostka polynomials in connection with the closure of nilpotent orbits. V = C n, G = GL(V ) N = {x End(V ) x : nilpotent } : nilpotent cone Closure relations : P n N /G λ G-orbit O λ x : Jordan type λ O λ = µ λ O µ (O λ : Zariski closure of O λ ) dominance order on P n For λ = (λ 1, λ 2,..., λ k ), µ = (µ 1, µ 2,..., µ k ), µ λ j i=1 µ i j i=1 λ i for each j. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 3 / 1

4 Notation: n(λ) = i 1 (i 1)λ i Define K λ,µ (t) = t n(µ) K λ,µ (t 1 ) : modified Kostka polynomial K = IC(O λ, C) : Intersection cohomology complex K : K i 1 d i 1 K i K = (K i ) : bounded complex of C-sheaves on O λ H i K = Ker d i / Im d i 1 : i-th cohomology sheaf d i Ki+1 d i+1 H i xk : the stalk at x O λ of H i K (finite dim. vecotr space over C) Known fact : H i K = 0 for odd i. Theorem (Lusztig) For x O µ, K λ,µ (t) = t n(λ) i 0(dim C H 2i x K)ti In particular, K λ,µ [t] Z 0 [t]. (theorem of Lascoux-Schützenberger) Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 4 / 1

5 Representation theory of GL n (F q ) F q : finite field of q elements with ch F q = p F q : algebraic closure of F q { } { 1 } G = GL n (F q ) B = U = B : Borel subgroup, U : maximal unipotent subgroup F : G G, (g ij ) (g q ij ) : Frobenius map G F = {g G F (g) = g} = G(F q ) : finite subgroup Ind G F B F 1 : the character of G F obtained by inducing up 1 B F Ind G F B F 1 = λ P n (deg χ λ )ρ λ, ρ λ : irreducible character of G F corresp. to χ λ S n P n. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 5 / 1

6 G uni = {g G u: unipotent} G, G uni N, u u 1 G uni /G P n, O λ λ O λ : F -stable = O F λ : single G F -orbit, u λ O F λ Theorem (Green) ρ λ (u µ ) = K λ,µ (q) Remark : Lusztig s result = the character values of ρ λ at unipotent elements are described in terms of intersection cohomology complex. Theory of character sheaves = describes all the character values of any irreducible characters in terms of certain simple perverse sheaves. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 6 / 1

7 Enhanced nilpotent cone λ = (λ (1),..., λ (r) ), P n,r : the set of r-partitions of n r i=1 λ(i) = n : r-partition of n S (2004) : for λ, µ P n,r, introduced K λ,µ (t) Q(t) : Kostka functions associated to complex reflection groups as the transition matrix between the bases of Schur functions {s λ (x)} and Hall-Littlewood functions {P µ (x; t)}. Achar-Henderson (2008) : geometric realization of Kostka functions for r = 2 V = C n, N : nilpotent cone N V : enhanced nilpotent cone, action of G = GL(V ) Achar-Henderson, Travkin : (N V )/G P n,2, O λ λ Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 7 / 1

8 K = IC(O λ, C) : Intersection cohomology complex Theorem (Achar-Henderson) H i K = 0 for odd i. For λ, µ P n,2, and (x, v) O µ O λ, t a(λ) i 0 (dim C H 2i (x,v) K)t2i = K λ,µ (t), where a(λ) = 2n(λ (1) ) + 2n(λ (2) ) + λ (2) for λ = (λ (1), λ (2) ). N V G uni V G V (over F q ) : action of G = GL(V ) Finkelberg-Ginzburg-Travkin (2008) : Theory of character sheaves on G V (certain G-equiv. simple perverse sheaves ) = character table of (G V ) F good basis of the space of G F -invariant functions on (G V ) F Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 8 / 1

9 Finite symmetric space GL 2n (F q )/Sp 2n (F q ) G = GL(V ) GL 2n (F q ), V = (F q ) 2n, ch F q ( 2 ) 0 θ : G G, θ(g) = J 1 ( t g 1 In )J : involution, J = I n 0 K := {g G θ(g) = g} Sp 2n (F q ) G F GL 2n (F q ) Sp 2n (F q ) K F G F acts on G F /K F 1 G F K F : induced representation G/K : symmetric space over F q H(G F, K F ) := End G F (1 G F K F ) : Hecke algebra asoc. to (G F, K F ) H(G F, K F ) : commutative algebra H(G F, K F ) : natural labeling by (GL F n ) K F \G F /K F : natural labeling by { conj. classes ofgl F n } Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 9 / 1

10 Theorem (Bannai-Kawanka-Song, 1990) The character table of H(G F, K F ) can be obtained from the character table of GL F n by replacing q q2. Geometric setting for G/K G ιθ = {g G θ(g) = g 1 } = {gθ(g) 1 g G}, where ι : G G, g g 1. The map G G, g gθ(g) 1 gives isom. G/K G ιθ. K acts by left mult G/K G ιθ K acts by conjugation. K\G/K {K-conjugates of G ιθ } Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 10 / 1

11 Henderson : Geometric reconstruction of BKS (not complete) Lie algerba analogue g = gl 2n, θ : g g : involution, g = g θ g θ, g ±θ = {x g θ(x) = ±x}, K-stable subspace of g g θ nil = g θ N g : analogue of nilpotent cone N, K-stable subset of g θ g θ nil /K P n, O λ λ Theorem (Henderson + BKS, 2008) Let K = IC(O λ, Q l ), x O µ O λ. Then H i K = 0 unless i 0 (mod 4), and t 2n(λ) (dim Hx 4i K)t2i = K λ,µ (t 2 ) i 0 Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 11 / 1

12 Exotic symmetric space GL 2n /Sp 2n V (Joint work with K. Sorlin) K λ,µ (t) (r = 1) Green, Lusztig GL n Bannai-Kawanaka-Song Henderson GL 2n /Sp 2n K λ,µ (t 2 ) (r = 1) K λ,µ (t 1/2 ) GL n V n (r = 2) Achar-Henderson Finkelberg-Ginzburg-Travkin GL2n /Sp 2n V 2n??? Kato s exotic nilcone Want to show K λ,µ (t) (r = 2) Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 12 / 1

13 G = GL(V ) GL 2n (F q ), dim V = 2n, K = G θ. X = G ιθ V : K action Problem Find a good class of K-equivariant simple perverse sheaves on G ιθ V, i.e., character sheaves on G ιθ V Find a good basis of K F -equivariant functions on (G ιθ V ) F, i.e., irreducible characters of (G ιθ V ) F, and compute their values, i.e., computaion of the character table Remark : X uni := G ιθ uni V g θ nil V : Kato s exotic nilcone Kato X uni /K (g θ nil V )/K P n,2, O µ µ P n,2 Natural bijection with GL n -orbits of enhanced nilcone, compatible with closure relations (Achar-Henderson) Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 13 / 1

14 Constrcution of character sheaves on X T B θ-stable maximal torus, θ-stable Borel sdubgroup of G M n : maximal isotropic subspace of V stable by B θ X = {(x, v, gb θ ) G ιθ V K/B θ g 1 xg B ιθ, g 1 x M n } π : X X, (x, v, gb θ ) (x, v), α : X T ιθ, (x, v, gb θ ) g 1 xg, (b b : projection B ιθ T ιθ ) T ιθ α X π X E : tame local system on T ιθ K T,E = π α E[dim X ] K T,E : semisimple perverse sheaf on X Definition X : (Character sheaves on X ) K-equiv. simple perverse sheaves on X, appearing as a direct summand of various K T,E. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 14 / 1

15 F q -structures on χ T,E and Green functions (T, E) as before Assume T : F -stable, (but B: not necessarily F -stable), F E E. Obtain canonical isomorphism ϕ : F K T,E K T,E Define a characteristic function χ K,ϕ of K = K T,E by χ K,ϕ (z) = i ( 1) i Tr (ϕ, H i zk) (z X F ) χ K,ϕ : K F -invariant function on X F Put χ T,E = χ KT,E,ϕ for each (T, E). Proposition-Definition χ T,E X F is independent of the choice of E on T ιθ. We define uni Q T : Xuni F Q l by Q T = χ T,E X F, and call it Green function on X F uni uni. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 15 / 1

16 Character formula For s G ιθ, semisimple, Z G (s) : θ-stable, and Z G (s) V has a similar structure as X = G ιθ V. Then Green function Q Z G (s) T (T : θ-stable maximal torus in Z G (s)) can be defined simialr to Q T Theorem (Character formula) Let s, u (G ιθ ) F be such that su = us, with s: semisimple, u: unipotent. Assume that E = E ϑ : F -stable tame local system on T ιθ with ϑ (T ιθ,f ). Then χ T,E (su, v) = Z K (s) F 1 x K F x 1 sx T ιθ,f Q Z G (s) xtx 1 (u, v)ϑ(x 1 sx) Remark The computation of the function χ T,E is reduced to the computation of Green functions Q Z G (s) xtx 1 for various semisimple s G ιθ. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 16 / 1

17 Orthogonality relations Theorem (Orthogonality relations for χ T,E ) Assume that T, T are F -stable, θ-stable maximal tori in G as before. Let E = E ϑ, E = E ϑ be tame local systems on T ιθ, T ιθ with ϑ (T ιθ,f ), ϑ (T ιθ,f ). Then K F 1 χ T,E (x, v)χ T,E (x, v) (x,v) X F = T θ,f 1 T θ,f 1 n N K (T θ,t θ ) F t T ιθ,f ϑ(t)ϑ (n 1 tn) Theorem(Orthogonality relations for Green functions) K F 1 (u,v) X F uni Q T (u, v) T (u, v) = N K (T θ, T θ ) F T θ,f T θ,f Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 17 / 1

18 Springer correspondence We consider the case E = Q l : constant sheaf on T ιθ. Then K T,E = π Q l [dim X ]. M 0 M 1 M n : isotorpic flag stable by B θ Define X m = g K g(bιθ M m ). Then X m : closed in X = X n, X 0 X 1 X n = X. W n = N K (T θ )/T θ : Weyl group of type C n, W n P n,2 Proposition 1 π Ql [dim X ] : a semisimple perverse sheaf with W n -action, is decomposed as π Ql [dim X ] V µ IC(X m(µ), L µ )[dim X m(µ) ], µ P n,2 V µ : standard irred. W n -module, m(µ) = µ (1) for µ = (µ (1), µ (2) ), L µ : local system on a smooth open subset of X m(µ). Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 18 / 1

19 Theorem (Springer correspondence) For each µ P n,2, IC(X m(µ), L µ ) Xuni IC(O µ, Q l ) (up to shift). Hence V µ O µ gives a bijective correspondence W n X uni/k. Remark 1 Springer correspondence was first proved by Kato for the exotic nilcone by using Ginzburg theroy on affine Hecke algebras. 2 The proof of the theorem is divided into two steps. In the first step, we show the existence of the bijection W n X uni/k. In the second step, we determine this map explcitly, by using an analogy of the restriction theorem due to Lusztig. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 19 / 1

20 Green functions and Springer correspondence We denote by T w F -stable, θ-stable maximal torus of G corresp. to w W n S 2n. For A µ = IC(O µ, Q l )[dim O µ ], we have a unique isomorphism ϕ µ : F A µ A µ induced from ϕ : F K T1, Q l K T1, Q l By using the Springer correspondence, we have Q Tw = ( 1) dim X dim X uni χ µ (w)χ Aµ,ϕ µ, µ P n,2 where χ µ is the irreducible characters of W n corresp. to V µ. Define, for each λ P n,2, Q λ = W n 1 w W n χ λ (w)q Tw Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 20 / 1

21 Then by using the orthogonality relations for Green functions, we have Proposition 2 For λ, µ P n,2, K F 1 Q λ (u, v)q µ (u, v) (u,v) X F uni = W n 1 Tw θ,f w W n 1 χ λ (w)χ µ (w). Remark By definition, we have for A λ = IC(O λ, Q l )[dim O λ ]. Q λ = ( 1) dim X dim X uni χ Aλ,ϕλ Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 21 / 1

22 Characterization of Kostka polynomials For any character χ of W n, we define n i=1 R(χ) = (t2i 1) W n w W n ε(w)χ(w) det V0 (t w), where ε : sign character of W n, and V 0 : reflection module of W n. R(χ) = graded multiplicity of χ in the coinvariant algerba R(W n ). Define a matrix Ω = (ω λ,µ ) λ,µ Pn,2 by ω λ,µ = t N R(χ λ χ µ ε). Define a partial order λ µ on P n,2 by the condition for any j; j i=0 j i=1 (λ (1) i + λ (2) i ) j i=1 j (λ (1) i + λ (2) i ) + λ (1) j+1 0=1 (µ (1) i + µ (2) i ) (µ (1) i + µ (2) i ) + µ (1) j+1. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 22 / 1

23 Theorem (S) There exists a unique matrices P = (p λ,µ ), Λ = (ξ λ,µ ) over Q[t] satisfying the equation PΛ t P = Ω subject to the condition that Λ is a diagonal matrix and { 0 unless µ λ, p λ,µ = t a(λ) if µ = λ. Then the entry p λ,µ coincides with K λ,µ (t). Remark. Under this setup, we have ω λ,µ (q) = K F W n 1 Tw θ,f 1 χ λ (w)χ µ (w). w W n Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 23 / 1

24 Conjecture of Achar-Henderson Main Theorem Let O λ be the orbit in X uni corresp. to λ P n,2, and put K = IC(O λ, Q l ). Then for (x, v) O µ O λ, we have H i K = 0 unless i 0 (mod 4), and t a(λ) i 0 (dim H 4i (x,v) K)t2i = K λ,µ (t). Remarks. 1 The theorem was first proved (for the exotic nilcone over C) by Kato by a different meothod. 2 By a similar argument, we can (re)prove a result of Henderson concerning the orbits in G ιθ g θ, without appealing the result from BKS. Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 24 / 1

25 K F -invariant functions on (G ιθ V ) F Let X be the set of character sheaves on X = G ιθ V. Put X F = {A X F A A}. For each A X F, fix an isomorphism ϕ A : F A A, and consider the characteristic function χ A,ϕA. Let C q (X ) be the Q l -space of K F -invariant functions on X F. Theorem 1 There exists an algorithm of computing χ A,ϕA for each A X F. 2 The set {χ A,ϕA A X F } gives a basis of C q (X ). Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 25 / 1