Kostka functions associated to complex reflection groups

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1 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

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 : 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

3 {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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 (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

15 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

16 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

17 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

18 Fix m = (m 1,..., m r 1 ) Z r 1 0 s.t. Put p i = m 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 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

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