Kostka functions associated to complex reflection groups

Kostka functions associated to complex reflection groups Toshiaki Shoji Tongji University March 25, 2017 Tokyo Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 1 / 30

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 : the set of partitions of n i λ i = n s λ (x) = s λ (x 1,..., x k ) Z[x 1,..., x k ] : Schur function s λ (x) = det(x λ j +k j i )/ det(x k j i ) P λ (x; t) = P λ (x 1,..., x k ; t) Z[x 1,..., x k ; t] : Hall-Littlewood function P λ (x; t) = ( w x λ 1 1 x ) λ x k i tx j k x i x j λ i >λ j w S k /S λ k Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 2 / 30

{s λ (x) λ P n }, {P λ (x; t) λ P n } : bases of the space (free Z[t]-module) of homog. symmetric functions of degee n For λ, µ P n, define Kostka polynomials K λ,µ (t) by s λ (x) = µ P n K λ,µ (t)p µ (x; t) (K λ,µ (t)) λ,µ Pn : Transition matrix of two bases {s λ (x)}, {P µ (x; t)} K λ,µ (t) Z[t] Notation: n(λ) = i 1 (i 1)λ i K λ,µ (t) = t n(µ) K λ,µ (t 1 ) : modified Kostka polynomials Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 3 / 30

Geometric realization of Kostka polynomials Lusztig (1981) : geometric realization of Kostka polynomials in connection with unipotent classes. V = C n, G = GL(V ) G uni = {x G x : unipotent} : unipotent variety P n G uni /G Dominance order of P n λ O λ x : Jordan type λ For partitions λ = (λ 1, λ 2,..., λ k ), µ = (µ 1, µ 2,..., µ k ) Closure relations : µ λ j i=1 µ i j i=1 λ i for each j O λ = µ λ O µ (O λ : Zariski closure of O λ ) Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 4 / 30

K = IC(O λ, C) : intersection cohomology complex K : K i 1 d i 1 K i d i Ki+1 d i+1 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 H i x K : stalk of H i K at x O λ (fin. dim. vector space ove C) Theorem (Lusztig 1981) For any odd i, we have H i K = 0. Moreover, for x O µ O λ, K λ,µ (t) = t n(λ) C Hx i 0(dim 2i K)t i In particular, K λ,µ [t] Z 0 [t]. (Theorem of Lascoux-Schützenberger) Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 5 / 30

Springer corresp. of GL n and Kostka polynomials G = GL n, B : Borel subgroup of G, T B : maximal torus, U B : maximal unipotent subgroup N G (T )/T S n : Weyl group of GL n π 1 : G uni = {(x, gb) G uni G/B g 1 xg U} G uni, (x, gb) x G uni : smooth, π 1 : proper surjective (Springer resolution of G uni ) Theorem (Lusztig, Borho-MacPherson) (π 1 ) C[dim G uni ] is a semisimple perverse sheaf on G uni, equipped with S n -action, and is decomposed as (π 1 ) C[dim G uni ] λ P n V λ IC(O λ, C)[dim O λ ], where V λ : irreducible S n -module corresp. to λ P n. Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 6 / 30

For x G uni, B x = {gb G/B g 1 xg B} π 1 1 (x) is called the Springer fibre of x. K = (π 1 ) C : complex with S n -action H i x K H i (π 1 1 (x), C) Hi (B x, C) as S n -modules S n -module H i (B x, C) : called the Springer module of S n Put d x = dim B x. H 2dx (B x, C) : cohomology of highest degree Corollary (Springer correps.) For x O µ, H 2dx (B x, C) V λ as S n -modules. By the corresp. x H 2dx (B x, C), obtain a natural bijection G uni /G S n. Proposition K λ,µ (t) = i n(λ) V λ, H 2i (B x, C) Sn t i (x O µ ) Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 7 / 30

Generalization of Kostka polynomials 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) Introduced Kostka functions K ± λ,µ (t) Q(t) (λ, µ P n,r ) assoc. to complex reflection group W n,r = S n (Z/rZ) n (K λ,µ (t)) :Transition matrix between basis {s λ (x)} of Schur functions and basis {P ± µ (x; t)} of Hall-Littlewood functions If r = 1, 2, K + λ,µ (t) = K λ,µ (t) Z[t]. (write as K λ,µ(t)) If r = 2, W n,2 : Weyl group of type B n (C n ). But those Kostka functins have no relations with Sp 2n or SO 2n+1. Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 8 / 30

Characterization of K ± λ,µ (t) For λ = λ (1),..., λ (r) ) P n,r, put a(λ) = r n(λ) + λ (2) + 2 λ (3) + + (r 1) λ (r) where n(λ) = n(λ (1) ) + + n(λ (r) ). K ± λ,µ (t) = ta(µ) K ± λ,µ (t 1 ) : modified Kostka function For λ = (λ (1),..., λ (r) ) P n,r, choose m 0 so that λ (k) = (λ (1) 1,..., λ(k) m ) for any k. Define a sequence c(λ) Z rm 0 by c(λ) = (λ (1) 1, λ(2) 1,..., λ(r) 1, λ(1) 2, λ(2) 2,..., λ(r) 2,..., λ(1) m, λ (2) m,..., λ (r) m ). Define a partial order µ λ in P n,r (dominance order on P n,r ) by c(µ) c(λ) under (gen. of ) the dominance order on P mr. Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 9 / 30

For any character χ of W n,r, define R(χ) by R(χ) = 1 i r (tir 1) W n,r w W n,r where V : reflection representation of W n,r. det V (w)χ(w) det V (t 1 V w) Fix a total order on P n,r compatible with, consider matrices indexed by P n,r with respect to.. Define a matrix Ω = (ω λ,µ ) λ,µ Pn,r by ω λ,µ = t N R(ρ λ ρ µ det V ) where ρ λ : irred. character of W n,r corresp. to λ P n,r, N : number of reflections in W n,r. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 10 / 30

Theorem (S 2004) There exist unique matrices P ±, Λ on Q(t) satisfying the relation P Λ t P + = Ω, where Λ : diagonal, P ± = (p ± λ,µ ) : lower triangular with p± λ,λ = ta(λ). Then p ± λ,µ = K ± λ,µ (t). Remarks. 1 Construction of K ± λ,µ depends on the choice of. Later we show independence of, and K ± λ,µ (t) Z[t]. 2 If r = 1, 2, W n,r : Weyl group, Ω : symmetric, so P + = P. If r 3, Ω : not symmetric, so P + P. 3 K λ,µ (t) have better properties than K + λ,µ (t) w.r.t geometry. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 11 / 30

Enhanced variety GL(V ) V Achar-Henderson (2008) : geometric realization of K λ,µ (t) in the case where r = 2. G = GL(V ), V = C n : n-dim. vector space G uni V G V : Enhanced variety, G acts diagonally (Achar-Henderson, Travkin) : (G uni V )/G P n,2, O λ λ Theorem (Achar-Henderson 2008) Put K = IC(O λ, C). If i is odd, then H i K = 0. 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). Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 12 / 30

Exotic symmetric space GL(V )/Sp(V ) V G = GL 2n (C) GL(V ), V : 2n-dim. vector space ( ) 0 θ : G G, θ(g) = J 1 ( t g 1 In )J : involution, J = I n 0 H := {g G θ(g) = g} Sp 2n (C) Define G/H : symmetric space 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/H G ιθ. X = G ιθ V X uni = G ιθ uni V : exotic symmetric space exotic nilpotent cone by Kato H acts diagonally on X and X uni. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 13 / 30

(Kato) X uni /H P n,2 (O λ λ) B = TU : θ-stable Borel subgroup, maximal torus, and maximal unipotent subgroup of G T θ B θ : maximal torus and Borel subgroup of H B = H/B θ : flag variety of H, N H (T θ )/T θ W n,2 Fix an isotropic flag in V stable by B θ M 1 M 2 M n V (dim M i = i) Put X uni = {(x, v, gb θ ) G ιθ uni V B g 1 xg B ιθ, g 1 v M n } and π 1 : Xuni X uni (x, v, gb θ ) (x, v). For z = (x, v) O λ, consider π 1 (z) X uni (exotic Springer fibre) π 1 (z) B z = {gb θ B g 1 xg B ιθ, g 1 v M n } Put d λ = (dim X uni dim O λ )/2 Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 14 / 30

Theorem (Kato, S.-Sorlin) (π 1 ) C[dim X uni ] is a semisimple perverse sheaf on X uni, equipped with W n,2 -action, and is decomposed as (π 1 ) C[dim X uni ] λ P n,2 Ṽ λ IC(O λ, C)[dim O λ ], where Ṽλ ; irred. representation of W n,2 corresp. to λ P n,2. Corollary (Springer corresp.) (Kato, S.-Sorlin) 1 H i (B z, C) has a structure of W n,2 -module (Springer module). 2 dim B z = d λ for z O λ. 3 H 2d λ(b z, C) Ṽλ as W n,2 -modules for z O λ. The corresp. O λ H 2d λ(b z, C) gives a natural bijection X uni /H W n,2 Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 15 / 30

Geometric realization of Kostka functions Theorem (Kato, S. -Sorlin) Put K = IC(O λ, C). Then for z O µ O λ, we have H i K = 0 unless i 0 (mod 4), and K λ,µ (t) = t a(λ) Hz i 0(dim 4i K)t 2i Corollary For z O µ, Kλ,µ (t) = i a(λ) H 2i (B z, C), Ṽλ Wn,2 t i. Remark. Springer correspondence can be formulated also for the enhanced variety. But in that case, W n,2 does not appear. Only the produc of symmetric groups S m S n m (0 m n) appears. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 16 / 30

Exotic symmeric space of higher level For an integer r 2, consider the varieties X = G ιθ V r 1 X uni = G ιθ uni V r 1 with diagonal action of H on X, X uni (exotic sym. space of level r). Remark. If r = 2, X = G ιθ V : exotic symmetric space. If r 3, X uni has infinitely many H-orbits. In fact, since dim G ιθ uni = 2n2 2n, Put dim X uni = 2n 2 2n + (r 1)2n > dim H = 2n 2 + n X uni = {(x, v, gb θ ) G ιθ uni V r 1 B g 1 xg B ιθ, g 1 v M r 1 n }, π 1 : Xuni X uni, (x, v, gb θ ) (x, v) : not surjective. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 17 / 30

Fix m = (m 1,..., m r 1 ) Z r 1 0 s.t. Put p i = m 1 + + m i for each i. Define X m,uni = g H X m,uni = π 1 1 (X m,uni), mi = n. r 1 g(u ιθ M pi ) X uni, i=1 let π m,1 : Xm,uni X m,uni : restriction of π 1, P(m) = {λ = (λ (1),..., λ (r) ) P n,r λ (i) = m i for 1 i r 2}. Theorem (S) (Springer correspondence for W n,r ) Let d m = dim X m,uni. Then (π m,1 ) C[d m] is a semisimple perverse sheaf on X m,uni equipped with W n,r -action, and is decomposed as (π m,1 ) C[d m] λ P(m) Ṽ λ IC(X λ, C)[dim X λ ], where Ṽλ : irred. rep. of W n,r corresp. to λ P(m) P n,r. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 18 / 30

Varieties X λ Note: m P(m) = P n,r For each λ P n,r, one can construct a locally closed, smooth, irreducible, H-stable subvariety X λ of X uni satisfying the following properties; 1 π 1 ( X uni ) = λ P n,r X λ. 2 X λ(m) = X m,uni for λ(m) = ((m 1 ), (m 2 ),..., (m r 1 ), ). 3 Assume that µ P(m). Then X µ X m,uni. 4 If r = 2, X λ coincides with H-orbit O λ. Remark. X λ : analogue of H-orbit O λ in the case r = 1, 2. However, if r 3, π 1 ( X uni ) λ P n,r X λ, and X λ are not mutually disjoint. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 19 / 30

Springer fibres For z = (x, v) X m,uni, put Hence B z : B z = π 1 1 (z) {gbθ B g 1 xg B ιθ, g 1 v M r 1 n }, B (m) z r 1 = {gb θ B z g 1 v M pi }.. B (m) z B z. called Springer fibre B (m) z : called small Springer fibre i=1 Remark. If r 3, dim B z, dim B z (m) for z X λ. are not necessarily constant Assume that λ P(m) (then X λ X m,uni ). Put d λ = (dim X m,uni dim X λ )/2. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 20 / 30

Proposition (Springer correspondence) Assume that λ P(m). 1 X 0 λ = {z X λ dim B (m) z = d λ } forms open dense subset of X λ. 2 Take z X 0 λ. Then H2d λ(b z, C) Ṽλ as W n,r -modules. The map X λ H 2d λ(b z, C) gives a bijective correspondence {X λ λ P n,r } W n,r Remarks. 1 In the case of GL(V ) V r 1 (enhanced variety of higher level), Springer correspondence can be proved. In that case, subgroups S m1 S mr S n with i m i = n appear. 2 Geometric realization of Kostka functions for r 3 is not yet known. Only exist partial results in case of enhanced vareity of higher level. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 21 / 30

Kostka functions with multi-variables y = (y 1, y 2,... ) : infinitely many variables Λ = Λ(y) = n 0 Λn : ring of symmetric functions w.r.t. y over Z. {s λ (y) λ P n } : free Z-basis of Λ n. Prepare r-types variables x = (x (1),..., x (r) ) with x (i) = (x (i) 1, x (i) 2,... ) : infinitely many variables Ξ = Ξ(x) = Λ(x (1) ) Λ(x (r) ) Ξ = n 0 Ξn : ring of symmetric functions w.r.t. x (1),..., x (r). For λ = (λ (1),..., λ (r) ) P n,r, define Schur function s λ (x) by s λ (x) = r s λ (k)(x (k) ). k=1 Then {s λ (x) λ P n,r } : free Z-basis of Ξ n. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 22 / 30

Z[t] = Z[t 1,..., t r ] : polynomial ring with r-variables t = (t 1,..., t r ). Ξ[t] = Z[t] Z Ξ = n 0 Ξn [t], Ξ(t) = Q(t) Z Ξ = n 0 Ξn Q (t) Fixing m 1, consider finitely many variables x (k) 1,..., x (k) m for each k. For a parameter t, define a function q (k) s,± (x; t) by q (k) s,± (x; t) = m (x (k) i i=1 ) s 1 (k) j 1 (x i tx (k 1) j ) (k) j i (x i x (k) j ) if s 1, and by q (k) s,± (x; t) = 1 if s = 0. (Here we regard k Z/rZ) Note : q (k) s,± (x; t) Z[x; t], symmetric w.r.t. x (k) and x (k 1). For λ P n,r, choose m 0 s.t. λ (k) = (λ (k) 1,..., λ(k) m ) for any k., Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 23 / 30

Define q ± λ (x; t) Z[x; t] by q ± λ (x; t) = m k Z/rZ i=1 q (k) λ (k) i,± (x; t k c), where c = 1 in the + -case, and c = 0 in the - -case. Bt taking m, obtain symmetric function q ± λ (x; t) Ξn [t]. {q ± λ (x; t) λ P n,r } : Q(t)-basis of Ξ n Q (t). We fix the total order on P n,r compatible with. Theorem (S, 2004, 2017) For any λ P n,r, there exists a unique function P ± λ (x; t) Ξn Q (t) satisfying the following properties; 1 P ± λ = µ λ c λ,µq ± µ with c λµ Q(t), and c λλ 0. 2 P ± λ = µ λ u λµs µ with u λµ Q(t) and u λλ = 1, Similarly, there exists a unique function Q ± λ (x; t) Ξn Q (t), defined by the condition c λλ = 1 in (i), and u λλ 0 in (ii). Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 24 / 30

Bt taking m, we obtain P ± λ (x; t), Q± λ (x; t) Ξn Q (t), called Hall-Littlewood functions with multi-parameter. By definition, P ± λ coincides with Q± λ, up to constant Q(t). {P ± λ λ P n,r }, {Q ± λ λ P n,r } give Q(t)-bases of Ξ n Q (t). For λ, µ P n,r, define K ± λ,µ (t) by s λ (x) = K ± λ,µ (t)p± µ (x; t). µ P n,r K ± λµ (t) = K ± λ,µ (t 1,..., t r ) : Kostka function with multi-variable Remarks. (i) When t 1 = = t r = t, P ± λ (x; t) coincides with P ± λ (x; t) introduced in (S, 2004). Hence in this case, K ± λ,µ (t,..., t) coincides with one variable K ± λ,µ (t) introduced there. (ii) In multi-variable case, no formula such as P Λ t P + = Ω. So the relation with complex reflection groups is not clear. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 25 / 30

Hall-Littlewood functions From the definition, P ± λ (x; t), Q± λ (x; t) are functions of x with coefficients in Q(t), and depend on the choice of on P n,r. Hence K ± λ,µ (t) Q(t), and may depend on the choice of. Theorem There exists a closed formula for P ± λ and Q± λ. In particular we have 1 P ± λ (x; t), Q± λ (x; t) Ξn [t]. 2 {P ± λ λ P n,r } gives a Z[t]-basis of Ξ n [t]. 3 K ± λ,µ (t) Z[t]. 4 In the defining formula of P ± λ in terms of q± µ or s µ, possible to replace by. Hence, P ± λ, K ± λ,µ are independent of the choice of. Remark. For the proof of the closed formula, we use, in an essential way, the argument based on the multi-parameter case. Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 26 / 30

Conjecture of Finkelberg-Ionov ε 1,..., ε m : standard basis of Z m, R + = {ε i ε j 1 i < j m} : positive roots of type A m 1 For any ξ = (ξ i ) Z m, define a polynomial L(ξ; t) Z[t] by L(ξ; t) = (m α) t m α, where (m α ) runs over all non-negative integers such that ξ = α R + m αα. L(ξ; t) : called Lusztig s partition function. Kostka polynomials and Lusztig s partition function For λ, µ P n, take m 0 s.t. λ = (λ 1,..., λ m ), µ = (µ 1,..., µ m ). Put δ 0 = (m 1, m 2,..., 1, 0). Then λ + δ 0, µ + δ 0 Z m. K λ,µ (t) = w S m ε(w)l(w 1 (λ + δ 0 ) (µ + δ 0 ); t) Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 27 / 30

Remark. Lusztig : gave q-analogue of weight multiplicity by affine Kazhdan-Lusztig polynomials Lusztig s conjecture : q-analogue of Kostant s weight multiplicity formula by Lusztig s partition function = proved by S.-I. Kato. Generalization to multi-variable Kostka functions Fix m 0, corresp. to λ P n,r, consider {x (k) i } {λ (k) i } (1 k r, 1 i m). Put M = {(k, i) 1 k r, 1 i m}. Put M = M = rm. Give a total order on M by (similarly as λ c(λ) Z M ) (1, 1) < (2, 1) < < (r, 1) < (1, 2) < (2, 2) < < (r, m). By using this order, identify Z M Z M Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 28 / 30

Let ε 1,..., ε M : standard basis of Z M, R + Z M : positive roots For ξ = (ξ i ) Z M, define L (ξ; t) by L (ξ; t) = (m α) (t b(ν) ) mα, (Lusztig s partition function) α where (m α ) runs over all non-negative integers satisfying (*): (*) ξ = α R + m αα with α = ε ν ε ν s.t. b(ν ) = b(ν) + 1. (Here b(ν) = k Z/rZ for ν = (k, i) M.) Theorem (conjecture of F-I) Put δ = (M 1, M 2,..., 1, 0). For λ, µ P n,r, we have K λ,µ (t) = w (S m) r ε(w)l (w 1 (c(λ) + δ) (c(µ) + δ); t). In particular, K λ,µ (t) is a monic of degree a(µ) a(λ). Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 29 / 30

Stability of Kostka functions Finkelberg-Ionov : geometric realization of K λ,µ (t) by coherent sheaf on a certain vector bundle on (GL m ) r (under condition below). Corollary Assume that µ (k) 1 > > µ (k) m > 0 for any k. Then K λ,µ (t) Z 0[t]. Let θ = (θ (1),..., θ (r) ) be r-partitions s.t. θ (k) = (θ 1,..., θ m ) (indep. of k). For λ, µ P n,r, we have λ + θ, µ + θ P n,r for some n. Proposition (stability) Let λ, µ P n,r, and assume that θ 1 θ 2 θ m > 0. Then K λ,µ (t) takes a value independent of the choice of θ. We have K λ+θ,µ+θ (t) = L (c(λ) c(µ); t). Kostka functions associated to complex reflectionmarch groups25, 2017 Tokyo 30 / 30