The Steinberg-Lusztig tensor product theorem, Casselman-Shalika and LLT polynomials
aa r X i v : . [ m a t h . R T ] A p r The Steinberg-Lusztig tensor product theorem,Casselman-Shalika and LLT polynomials
Martina Lanini email: [email protected] Ram email: [email protected]
Dedicated to Friedrich Knop and Peter Littelmannon the occasion of their 60th birthdays
Abstract
In this paper we establish a Steinberg-Lusztig tensor product theorem for abstract Fockspace. This is a generalization of the type A result of Leclerc-Thibon and a Grothendieckgroup version of the Steinberg-Lusztig tensor product theorem for representations of quantumgroups at roots of unity. Although the statement can be phrased in terms of parabolic affineKazhdan-Lusztig polynomials and thus has geometric content, our proof is combinatorial,using the theory of crystals (Littelmann paths). We derive the Casselman-Shalika formulaas a consequence of the Steinberg-Lusztig tensor product theorem for abstract Fock space.
Key words— quantum groups, affine Lie algebras, Hecke algebras, symmetric functions
In our previous paper [LRS] we provided a construction of an “abstract” Fock space F ℓ ina general Lie type setting. The construction is given by simple combinatorial “straighteningrelations” which generalize the Kashiwara-Miwa-Stern [KMS] formulation of the q -Fock spacefrom the type A case. We showed that the abstract Fock space is a combinatorial realization ofthe graded Grothendieck group of finite dimensional representations of the quantum group at aroot of unity, where the standard basis elements | λ i correspond to the Weyl modules ∆ q ( λ ) andthe KL-basis C λ corresponds to the simple modules L q ( λ ).In Section 1 we prove a product theorem (Theorem 1.3) in abstract Fock space which gen-eralizes the type A theorem of Leclerc and Thibon [LT, Theorem 6.9]. Our proof follows thesame pattern as the proof for Type A given in [LT, Theorem 6.9] except that, in order to dealwith general Lie type, we have replaced the use of ribbon tableaux with the crystal basis andLittelmann paths. The basic philosophy of our technique is similar to the main idea of a paperof Guilhot [Gu] but we also make use of the elegant cancellation technique of Littelmann [Li,proof of Theorem 9.1] to complete the proof. This technique provides a combinatorial controlof the Demazure operator used in the proof of [Knp, Lemma 4.4]. We have not consideredthe unequal parameter case in this paper but the close relation between our context and thatof [Knp] cries out for an extension of the tensor product theorem for abstract Fock space tounequal parameters.The Casselman-Shalika formula is important in the representation theory of p-adic groups(see [CS]), in its relation to the affine Hecke algebra (see for example [BBF]) and in the geometric AMS Subject Classifications: Primary 17B37; Secondary 20C20. t .Kazhdan and Lusztig [KL94] established an equivalence of categories between an appropriatecategory of representations of the affine Lie algebra (of negative level) and the finite dimensionalrepresentations of the quantum group (of the finite dimensional Lie algebra) at a root of unity.In Section 4 we review this correspondence and make explicit the tensor product theorem interms of representations of the affine Lie algebra. This produces a character formula for certainnegative level irreducible highest weight representations of the affine Lie algebra. From the pointof view of this paper this character formula is an easy consequence of [KL94] and [Lu89]. Wefind it difficult to believe that this formula has not been noticed before but we have not yet beenable to locate a suitable specific reference.We thank all the institutions which have supported our work on this paper, particularly theUniv. of Melbourne, the Australian Research Council (grants DP1201001942 and DP130100674)and ICERM (Institute for Computational and Experimental Research in Mathematics). Wethank Kari Vilonen and Ting Xue for generous support of a visit of Martina Lanini to Universityof Melbourne funded by their Australian Research Council grant DP150103525. F ℓ Let W be a finite Weyl group, generated by simple reflections s , . . . , s n , and acting on a latticeof weights a ∗ Z . For example, this situation arises when T is a maximal torus of a reductivealgebraic group G , a ∗ Z = Hom( T, C × ) and W = N ( T ) /T, (1.1)where N ( T ) is the normalizer of T in G . The simple reflections in W correspond to a choice ofBorel subgroup B of G which contains T . Let R + denote the positive roots. Let α , . . . , α n bethe simple roots and let α ∨ , . . . , α ∨ n be the simple coroots. The dot action of W on a ∗ Z is given2teinberg product theorem in abstract Fock Spaceby w ◦ λ = w ( λ + ρ ) − ρ, where ρ = X α ∈ R + α (1.2)is the half sum of the positive roots for G (with respect to B ).Fix ℓ ∈ Z > . The abstract Fock space F ℓ is the Z [ t , t − ]-module generated by {| λ i | λ ∈ a ∗ Z } with relations | s i ◦ λ i = −| λ i , if h λ + ρ, α ∨ i i ∈ ℓ Z ≥ , − t | λ i , if 0 < h λ + ρ, α ∨ i i < ℓ , − t | s i ◦ λ (1) i − | λ (1) i − t | λ i , if h λ + ρ, α ∨ i i > ℓ and h λ + ρ, α ∨ i i 6∈ ℓ Z , (1.3)where λ (1) = λ − jα i if h λ + ρ, α ∨ i i = kℓ + j with k ∈ Z > and j ∈ { , . . . , ℓ − } .The following picture illustrates the terms in (1.3). This is the case G = SL with ℓ = 5, h ω , α ∨ i = 1 and α = 2 ω and, in the picture, λ corresponds to the third case of (1.3), µ to thefirst case and ν to the second case. s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s − − − − − − − − − − − − − − ρ ω λλ (1) s ◦ λ (1) s ◦ λ µs ◦ µ νs ◦ ν Define a Z -linear involution : F ℓ → F ℓ by t = t − and | λ i = ( − ℓ ( w ) ( t − ) ℓ ( w ) − N λ | w ◦ λ i . (1.4)where w is the longest element of W , ℓ ( w ) = Card( R + ) is the length of w , and N λ =Card { α ∈ R + | h λ + ρ, α ∨ i ∈ ℓ Z } . The dominant integral weights with the dominance partial order ≤ are the elements of( a ∗ Z ) + = { λ ∈ a ∗ Z | h λ + ρ, α ∨ i i > i = 1 , , . . . , n } with µ ≤ λ if µ ∈ λ − P α ∈ R + Z ≥ α. (1.5)In [LRS, Theorem 1.1 and Proposition 2.1] we showed that F ℓ has bases {| λ i | λ ∈ ( a ∗ Z ) + } and { C λ | λ ∈ ( a ∗ Z ) + } (1.6)where C λ are determined by C λ = C λ and C λ = | λ i + X µ = λ p µλ | µ i , with p µλ ∈ t Z [ t ]. (1.7) K [ X ] W on F ℓ Letting K = Z [ t , t − ], the group algebra of a ∗ Z is K [ X ] = K -span { X µ | µ ∈ a ∗ Z } with X µ X ν = X µ + ν . (1.8)The Weyl group W acts K -linearly on K [ X ] by wX µ = X wµ , for w ∈ W and µ ∈ a ∗ Z , and K [ X ] W = { f ∈ K [ X ] | wf = f } (1.9)3teinberg product theorem in abstract Fock Spaceis the ring of symmetric functions .Let V be the free K -module generated by {| λ i | λ ∈ a ∗ Z } so that F ℓ ∼ = V /I, (1.10)where I is the subspace of V consisting of K -linear combinations of the elements a λ = | s i ◦ λ i + | λ i , with h λ + ρ, α ∨ i i ∈ ℓ Z ≥ , b λ = | s i ◦ λ i + t | λ i , with 0 < h λ + ρ, α ∨ i i < ℓ , and c λ = | s i ◦ λ i + t | s i ◦ λ (1) i + | λ (1) i + t | λ i , with h λ + ρ, α ∨ i i > ℓ and h λ + ρ, α ∨ i i 6∈ ℓ Z .Let ˚ g be the Lie algebra of the reductive group G alluded to in (1.1). Let ϕ be the highestweight of the adjoint representation and let ϕ ∨ ∈ [˚ g ϕ , ˚ g − ϕ ] such that h ϕ, ϕ ∨ i = 2 (so that ϕ ∨ isan appropriate normalized highest short coroot of ˚ g ).The dual Coxeter number is h = h ρ, ϕ ∨ i + 1 . (1.11)The level ( − ℓ − h ) action of K [ X ] on V is the K -linear extension of X µ · | γ i = | − ℓw µ + γ i , for µ, γ ∈ a ∗ Z . (1.12)Letting w be the longest element of W , define µ ∗ = − w µ and w ∗ = w ww , for µ ∈ a ∗ Z and w ∈ W .(This notation is such that if µ ∈ a ∗ Z and L ˚ g ( µ ) denotes the irreducible ˚ g -module of highestweight µ then the dual L ˚ g ( µ ) ∗ ∼ = L ˚ g ( µ ∗ ) and ∗ : W → W is the involutive automorphism of W induced by the automorphism of the Dynkin diagram specified by s ∗ i = s i ∗ .) Then X µ · | γ i = | ℓµ ∗ + γ i and ( wµ ) ∗ = w ∗ µ ∗ . (1.13)The following proposition establishes an action of the ring of symmetric functions K [ X ] W on the abstract Fock space F ℓ . From the point of view of Theorem 2.2 below, this action iscoming from an action of the center of the affine Hecke algebra which, by an important resultof Bernstein, is the ring of symmetric functions (inside the affine Hecke algebra). Our proof ofProposition 1.1 provides an independent proof of the existence of the action of K [ X ] W withoutreferring to the affine Hecke algebra and the characterization of its center. Proposition 1.1.
The action of K [ X ] on V given in (1.12) induces a K -linear action of thering K [ X ] W of symmetric functions on F ℓ by X w ∈ W X wµ · | γ i = X w ∈ W | ℓ ( wµ ) ∗ + γ i , for µ ∈ a ∗ Z and γ ∈ a ∗ Z .Proof. Let f be an element of the subspace I defined in (1.10), let µ ∈ a ∗ Z and let i ∈ { , . . . , n } .Summing over a set of representatives of the cosets in { , s ∗ i }\ W , X w ∈ W X wµ · f = X v ∈{ ,s ∗ i }\ W ( X vµ + X s ∗ i vµ ) · f, where the representatives v ∈ { , s ∗ i }\ W are chosen such that h vµ, α ∨ i ∗ i ∈ Z ≥ .4teinberg product theorem in abstract Fock SpaceCase 1: f = | s i ◦ λ i + | λ i with h λ + ρ, α ∨ i i ∈ ℓ Z ≥ . Then( X s ∗ i vµ + X vµ ) · ( | s i ◦ λ i + | λ i )= | ℓ ( s ∗ i vµ ) ∗ + s i ◦ λ i + | ℓ ( vµ ) ∗ + s i ◦ λ i + | ℓ ( s ∗ i vµ ) ∗ + λ i + | ℓ ( vµ ) ∗ + λ i = | ℓs i v ∗ µ ∗ + s i ◦ λ i + | ℓv ∗ µ ∗ + s i ◦ λ i + | ℓs i v ∗ µ ∗ + λ i + | ℓv ∗ µ ∗ + λ i = | s i ◦ ( ℓv ∗ µ ∗ + λ ) i + | s i ◦ ( ℓs i v ∗ µ ∗ + λ ) i + | ℓs i v ∗ µ ∗ + λ i + | ℓv ∗ µ ∗ + λ i = ( a ℓv ∗ µ ∗ + s i ◦ λ + a ℓv ∗ µ ∗ + λ , if h ℓv ∗ µ ∗ , α ∨ i i > h λ + ρ, α ∨ i i , a ℓs i v ∗ µ ∗ + λ + a ℓv ∗ µ ∗ + λ , if h ℓv ∗ µ ∗ , α ∨ i i ≤ h λ + ρ, α ∨ i i .Thus the right hand side is an element of I .Case 2: f = | s i ◦ λ i + t | λ i with 0 < h λ + ρ, α ∨ i i < ℓ . Then ℓv ∗ µ ∗ + s i ◦ λ = ( ℓv ∗ µ ∗ + λ ) (1) sothat( X s ∗ i vµ + X vµ ) · ( | s i ◦ λ i + t | λ i )= | ℓ ( s ∗ i vµ ) ∗ + s i ◦ λ i + | ℓ ( vµ ) ∗ + s i ◦ λ i + t | ℓ ( s ∗ i vµ ) ∗ + λ i + t | ℓ ( vµ ) ∗ + λ i = | ℓs i v ∗ µ ∗ + s i ◦ λ i + | ℓv ∗ µ ∗ + s i ◦ λ i + t | ℓs i v ∗ µ ∗ + λ i + t | ℓv ∗ µ ∗ + λ i = | s i ◦ ( ℓv ∗ µ ∗ + λ ) i + | ( ℓv ∗ µ ∗ + λ ) (1) i + t | s i ◦ ( ℓv ∗ µ ∗ + s i ◦ λ ) i + t | ℓv ∗ µ ∗ + λ i = | s i ◦ ( ℓv ∗ µ ∗ + λ ) i + t | s i ◦ ( ℓv ∗ µ ∗ + λ ) (1) i + | ( ℓv ∗ µ ∗ + λ ) (1) i + t | ℓv ∗ µ ∗ + λ i = ( c ℓv ∗ µ ∗ + λ , if h v ∗ µ ∗ , α ∨ i i ∈ Z > ,2 b ℓv ∗ µ ∗ + λ , if h v ∗ µ ∗ , α ∨ i i = 0,since if s ∗ i vµ = vµ then s i v ∗ µ ∗ = v ∗ µ ∗ and h v ∗ µ ∗ , α ∨ i i ∈ Z > then h ℓv ∗ µ ∗ + λ + ρ, α ∨ i i > ℓ and h ℓv ∗ µ ∗ + λ + ρ, α ∨ i i 6∈ ℓ Z . Thus the right hand side is an element of I .Case 3: Assume λ ∈ a ∗ Z with h λ + ρ, α ∨ i i > ℓ and h λ + ρ, α ∨ i i 6∈ ℓ Z . If µ ∈ a ∗ Z and h ν, α ∗ i i ∈ Z ≥ then s i ◦ ( v ∗ µ ∗ + ν ) = s i ( v ∗ µ ∗ + ν + ρ ) − ρ = s i v ∗ µ ∗ + s i ◦ ν and ( ℓν ∗ + λ ) (1) = ℓν ∗ + λ (1) , so that, with h λ + ρ, α ∨ i i = kℓ + j with k > ≤ j < ℓ ,( X s ∗ i vµ + X vµ ) · ( | s i ◦ λ i + t | s i ◦ λ (1) i + | λ (1) i + t | λ i )= | ℓ ( s ∗ i vµ ) ∗ + s i ◦ λ i + | ℓ ( vµ ) ∗ + s i ◦ λ i + t | ℓ ( s ∗ i vµ ) ∗ + s i ◦ λ (1) i + t | ℓ ( vµ ) ∗ + s i ◦ λ (1) i + | ℓ ( s ∗ i vµ ) ∗ + λ (1) i + | ℓ ( vµ ) ∗ + λ (1) i + t | ℓ ( s ∗ i vµ ) ∗ + λ i + t | ℓ ( vµ ) ∗ + λ i = | ℓs i v ∗ µ ∗ + s i ◦ λ i + | ℓv ∗ µ ∗ + s i ◦ λ i + t | ℓs i v ∗ µ ∗ + s i ◦ λ (1) i + t | ℓv ∗ µ ∗ + s i ◦ λ (1) i + | ℓs i v ∗ µ ∗ + λ (1) i + | ℓv ∗ µ ∗ + λ (1) i + t | ℓs i v ∗ µ ∗ + λ i + t | ℓv ∗ µ ∗ + λ i = | s i ◦ ( ℓv ∗ µ ∗ + λ ) i + | s i ◦ ( ℓs i v ∗ µ ∗ + λ ) i + t | s i ◦ ( ℓv ∗ µ ∗ + λ (1) ) i + t | s i ◦ ( ℓs i v ∗ µ ∗ + λ (1) ) i + | ℓs i v ∗ µ ∗ + λ (1) i + | ( ℓv ∗ µ ∗ + λ ) (1) i + t | ℓs i v ∗ µ ∗ + λ i + t | ℓv ∗ µ ∗ + λ i = c λ + ℓv ∗ µ ∗ + c s i ◦ λ (1) + v ∗ µ ∗ , if h ℓv ∗ µ ∗ , α ∨ i i > ℓk > c λ + ℓv ∗ µ ∗ + c λ + s i v ∗ µ ∗ , if 0 < h ℓv ∗ µ ∗ , α ∨ i i < ℓk , c λ + ℓv ∗ µ ∗ + b λ + s i v ∗ µ ∗ + b s i ◦ λ (1) + v ∗ µ ∗ , if h ℓv ∗ µ ∗ , α ∨ i i = ℓk .Thus the right hand side is an element of I .These computations show that I is stable under the action of K [ X ] W . Thus the action of K [ X ] W on F ℓ = V /I is well defined. 5teinberg product theorem in abstract Fock Space
Remark 1.2.
One might be tempted to try to define an action of K [ X ] on F ℓ by X µ · | γ i = | γ + ℓµ i for µ, γ ∈ a ∗ Z but this action is not well defined. For example in the G = SL casewith ℓ = 5 pictured after (1.3), one would have 0 = X ω · | − i = | − i = | i , which is acontradiction to (1.6). On the other hand 0 = ( X − ω + X ω ) · | − i = | − − i + | i = 0, as itshould be. Let ˚ g be the Lie algebra of the reductive group G alluded to in (1.1). For λ ∈ ( a ∗ Z ) + let L ˚ g ( λ )be the irreducible U ˚ g -module of highest weight λ and let B ( λ ) be the crystal of L ˚ g ( λ ), B ( λ ) = { LS paths p of type λ } and wt( p ) denotes the endpoint of p ,see [Ra, § Weyl character corresponding to λ is the element of K [ X ] W given by s λ = char( L ˚ g ( λ )) = X w ∈ W det( w ) X w ◦ λ X w ∈ W det( w ) X w ◦ = X p ∈ B ( λ ) X wt( p ) . (1.14)An ℓ -restricted dominant integral weight is λ ∈ ( a ∗ Z ) + such that h λ , α ∨ i i < ℓ for i ∈{ , . . . , n } . In other words, if ω , . . . , ω n are the fundamental weights for ˚ g then a weight λ ∈ ( a ∗ Z ) + is ℓ -restricted if λ is an element ofΠ ℓ = { a ω + · · · + a n ω n | a , . . . , a n ∈ { , , . . . , ℓ − }} . (1.15) Theorem 1.3.
Let λ ∈ ( a ∗ Z ) + be a dominant integral weight and write λ = ℓλ + λ , with λ ∈ Π ℓ and λ ∈ ( a ∗ Z ) + .Then, with C λ ∈ F ℓ as in (1.7) and the K [ X ] W -action on F ℓ as in Proposition 1.1, C λ = s λ ∗ · C λ . Proof.
The proof is accomplished in two steps:(a) Show that s λ ∗ · C λ is bar invariant.(b) Show that s λ ∗ · C λ = | λ i + P µ = λ c µ | µ i with c µ ∈ t Z [ t ].Proof of (a): The bar involution and N γ are defined in (1.4). Since h− ℓw µ, α ∨ i ∈ ℓ Z then N γ − ℓw µ = Card { α ∈ R + | h γ − ℓw µ + ρ, α ∨ i ∈ ℓ Z } = Card { α ∈ R + | h γ + ρ, α ∨ i ∈ ℓ Z } = N γ . Thus X µ · | γ i = | γ − ℓw µ i = ( − ℓ ( w ) ( t − ) ℓ ( w ) − N γ − ℓw µ | w ◦ ( γ − ℓw µ ) i = ( − ℓ ( w ) ( t − ) ℓ ( w ) − N γ − ℓw µ | w ( γ + ρ ) − ρ − ℓµ i = ( − ℓ ( w ) ( t − ) ℓ ( w ) − N γ − ℓw µ | w ◦ γ − ℓµ i = ( − ℓ ( w ) ( t − ) ℓ ( w ) − N γ | w ◦ γ − ℓµ i = ( − ℓ ( w ) ( t − ) ℓ ( w ) − N γ X w µ · | w ◦ γ i = X w µ · | γ i , s λ ∗ is W -invariant, s λ ∗ · C λ = ( w s λ ∗ ) · C λ = s λ ∗ · C λ . (b) Let λ = λ + ℓλ as in the statement of the Theorem and let a ≡ b mean a = b mod t . Bythe second formula in (1.7), s λ ∗ · C λ ≡ s λ ∗ · | λ i = X p ∈ B ( λ ∗ ) X wt( p ) | λ i = X p ∈ B ( λ ∗ ) | ℓ wt( p ) ∗ + λ i . (1.16)By (1.3), if λ ∈ a ∗ Z and h λ + ρ, α ∨ i i ≥ | s i ◦ λ i ≡ −| λ i , if h λ + ρ, α ∨ i i ∈ ℓ Z ≥ ,0 , if 0 < h λ + ρ, α ∨ i i < ℓ , −| λ (1) i , otherwise,where λ (1) = λ − jα i if h λ + ρ, α ∨ i i = kℓ + j with j ∈ { , , . . . , ℓ − } . Since λ (1) = λ if h λ + ρ, α ∨ i i ∈ ℓ Z ≥ , the first case can be viewed as a special case of the last case to read | s i ◦ λ i ≡ ( , if 0 < h λ + ρ, α ∨ i i < ℓ , −| λ (1) i , otherwise.Assume h ν + ρ, α ∨ i i ∈ Z ≤ and let λ = s i ◦ ( λ + ℓν ). Since h ρ, α ∨ i i = 1 then h λ + ρ, α ∨ i i = h s i ◦ ( λ + ℓν ) + ρ, α ∨ i i = h s i ( λ + ℓν + ρ ) , α ∨ i i = h λ + ℓν + ρ, s i α ∨ i i = −h λ + ℓν + ρ, α ∨ i i = ℓ ( −h ν + ρ, α ∨ i i ) + ( ℓ − − h λ , α ∨ i i ) . Since λ ∈ Π ℓ then 0 ≤ ℓ − − h λ , α ∨ i i < ℓ and so λ (1) = λ − ( ℓ − − h λ , α ∨ i i ) α i = s i ◦ ( λ + ℓν ) − ( ℓ − − h λ , α ∨ i i ) α i = s i λ + ℓs i ν + s i ρ − ρ − ( ℓ − − h λ , α ∨ i i ) α i = ( λ − h λ , α ∨ i i α i ) + ℓ ( s i ν + s i ρ − ρ ) + ( ℓ − α i − ( ℓ − − h λ , α ∨ i i ) α i = λ + ℓ ( s i ◦ ν ) . Thus, since s i ◦ λ = λ + ℓν , | λ + ℓν i ≡ ( , if h ν + ρ, α ∨ i i = 0, −| λ + ℓ ( s i ◦ ν ) i , if h ν + ρ, α ∨ i i < s i ◦ ν = ν when h ν + ρ, α ∨ i i = 0, then | λ + ℓν i ≡ −| λ + ℓ ( s i ◦ ν ) i when h ν + ρ, α ∨ i i ∈ Z ≤ and, replacing ν by s i ◦ ν , gives | λ + ℓν i ≡ −| λ + ℓ ( s i ◦ ν ) i for h ν + ρ, α ∨ i i ∈ Z ≥ . Thus | λ + ℓν i ≡ −| λ + ℓ ( s i ◦ ν ) i , for ν ∈ a Z . (1.17)With formula (1.17) established, follow [Ra, proof of Theorem 5.5] (see also [Li, proof ofTheorem 9.1]) to define an involution ι on the set B ( λ ∗ ) \ { p + λ ∗ } , where p + λ ∗ is the unique highestweight path in B ( λ ∗ ).Let p ∈ B ( λ ∗ ) and p = p + λ ∗ . Since p = p + λ ∗ the path p crosses a wall out of the fundamentalchamber at some point during its trajectory. Let r be such that the first time p leaves the7teinberg product theorem in abstract Fock Spacedominant chamber is by crossing the hyperplane { x ∈ a ∗ R | h x, α ∨ r i = 0 } . Letting ˜ e r and ˜ f r denote the root operators on B ( λ ∗ ), the r -string containing p is S r ( p ) = { q ∈ B ( λ ∗ ) | q = ˜ e kr p or q = ˜ f kr p where k ∈ Z ≥ } . Let ι ( p ) be the element of S r ( p ) such that wt( ι ( p )) = s r ◦ wt( p ).By (1.17), | λ + ℓ wt( p ) ∗ i ≡ −| λ + ℓ ( s ∗ r ◦ wt( p ) ∗ ) i = −| λ + ℓ ( s r ◦ wt( p )) ∗ i = −| λ + ℓ wt( ι ( p )) ∗ i , and so the map ι partitions the set B ( λ ∗ ) \ { p + λ ∗ } into pairs { p, ι ( p ) } which cancel each other inthe mod t straightening of the terms of s λ ∗ · C λ in (1.16). Thus s λ ∗ · C λ ≡ | ℓ ( λ ∗ ) ∗ + λ i = | λ + ℓλ i , which proves (b). In order to establish the Casselman-Shalika formula it is necessary to use the connection betweenthe abstract Fock space F ℓ and the affine Hecke algebra H . Let us recall this relationship from[LRS]. H Keep the notation for the finite Weyl group W , the simple reflections s , . . . , s n and the weightlattice a ∗ Z as in (1.1). For i, j ∈ { , . . . , n } with i = j , let m ij denote the order of s i s j in W so that s i = 1 and ( s i s j ) m ij = 1 are the relations for the Coxeter presentation of W . Let K = Z [ t , t − ]. The affine Hecke algebra is H = K -span { X µ T w | µ ∈ a ∗ Z , w ∈ W } , (2.1)with K -basis { X µ T w | µ ∈ a ∗ Z , w ∈ W } and relations( T s i − t )( T s i + t − ) = 0 , T s i T s j T s i . . . | {z } m ij factors = T s j T s i T s j . . . | {z } m ij factors , (2.2) X λ + µ = X λ X µ , and T s i X λ − X s i λ T s i = ( t − t − ) (cid:18) X λ − X s i λ − X − α i (cid:19) , (2.3)for i, j ∈ { , . . . , n } with i = j and λ, µ ∈ a ∗ Z . The bar involution on H is the Z -linear automor-phism : H → H given by t = t − , T s i = T − s i , and X λ = T w X w λ T − w . (2.4)for i = 1 , . . . , n and λ, µ ∈ a ∗ Z . For µ ∈ a ∗ Z and w ∈ W define X t µ w = X µ ( T w − ) − and T t µ w = T x X µ + T w µ + ( T w − xw µ + ) − , (2.5)8teinberg product theorem in abstract Fock Spacewhere µ + is the dominant representative of W µ , x ∈ W is minimal length such that µ = xµ + and w µ + is the longest element of the stabilizer W µ + = Stab W ( µ +). Define ε = ( − t ) ℓ ( w ) X z ∈ W ( − t − ) ℓ ( z ) T z and = ( t − ) ℓ ( w ν ) X z ∈ W ( t ) ℓ ( z ) T z , so that ε = ε , = , and ε T s i = − t − ε , and T s i = t , (2.6)for i ∈ { , . . . , n } . The algebra K [ X ] defined in (1.8) is a subalgebra of H and, by a theorem ofBernstein (see [NR, Theorem 1.4]), the center of H is the ring of symmetric functions, Z ( H ) = K [ X ] W . (2.7) Remark 2.1.
Formulas (2.4) and (2.5) are just a reformulation of the usual bar involution andthe conversion between the Bernstein and Coxeter presentations of the affine Hecke algebra (seefor example [NR, Lemma 2.8 and (1.22)]). H and the abstract Fock space F ℓ In this subsection we follow [LRS, § affine Weyl group is W = { t µ w | µ ∈ a ∗ Z , w ∈ W } , with t µ t ν = t µ + ν , and wt µ = t wµ w, (2.8)for µ, ν ∈ a ∗ Z and w ∈ W . Let ϕ ∨ and h be as in (1.11). For ℓ ∈ Z > , the level ( − ℓ − h ) dotaction of W on a ∗ Z is given by( t µ w ) ◦ λ = ( w ◦ λ ) − ℓµ = w ( λ + ρ ) − ρ − ℓµ, (2.9)for µ ∈ a ∗ Z , w ∈ W and λ ∈ a ∗ Z . Note that this is an extension of the dot action of W given in(1.2). Define A − ℓ − h = { ν ∈ a ∗ Z | h ν, ϕ ∨ i ≥ − ℓ − h ν, α ∨ i i ≤ − i ∈ { , . . . , n }} . (2.10)and P + − ℓ − h = M ν ∈ A − ℓ − h ε H p ν , (2.11)where ε is as in (2.6) and p ν are formal symbols indexed by ν ∈ A − ℓ − h satisfying p ν = p ν and T y p ν = ( t ) ℓ ( y ) p ν for y ∈ W ν ,where W ν = Stab W ( ν ) is the stabilizer of ν under the level ( − ℓ − h ) dot action of W on a ∗ Z .Define a bar involution: P + − ℓ − h → P + − ℓ − h by ε f p ν = ε ¯ f p ν , for ν ∈ A − ℓ − h and f ∈ H . (2.12)For λ ∈ a ∗ Z define[ X λ ] = [ X w v ◦ ν ] = ε X v p ν , where λ = w v ◦ ν with ν ∈ A − ℓ − h , (2.13)and v ∈ W is such that X vu = X v T u for any u ∈ W ν . It is helpful to stress that the ( − ℓ − h )dot action of (2.9) applies here so that, when v = t µ w with µ ∈ a ∗ Z and w ∈ W , then λ = − ℓw µ + ( w w ) ◦ ν and[ X λ ] = [ X − ℓw µ +( w w ) ◦ ν ] = ε X t µ w p ν = ε X µ ( T w − ) − p ν . (2.14)With these notations, a main result of [LRS] is9teinberg product theorem in abstract Fock Space Theorem 2.2. (see [LRS, Theorem 4.7]) Let ≤ be the dominance order on the set ( a ∗ Z ) + ofdominant integral weights. Then the K -linear map Φ : F ℓ → P + − ℓ − h given by Φ( | λ i ) = [ X λ ] , for λ ∈ a ∗ Z , (2.15) is a well defined K -module isomorphism satisfying Φ( f ) = Φ( f ) . Since elements of Z ( H ) = K [ X ] W commute with ε there is a K [ X ] W -action on P − ℓ − h by leftmultiplication. The pullback of this action by the isomorphism Φ is the source of the K [ X ] W action on F ℓ given in Proposition 1.1, z Φ( f ) = Φ( zf ) , for z ∈ Z ( H ) = K [ X ] W and f ∈ F ℓ . (2.16) For µ ∈ a ∗ Z define the “Whittaker function” A µ ∈ ε H by A µ = ε X µ . (2.17)See, for example, [HKP, §
6] for the connection between p -adic groups and the affine Heckealgebra and the explanation of why A µ is equivalent to the data of a (spherical) Whittakerfunction for a p -adic group. As proved carefully in [NR, Theorem 2.7], it follows from (2.6) and(2.3) that ε H has K -basis { A λ + ρ | h λ + ρ, α i i ∈ Z ≥ for i ∈ { , . . . , n }} . Following [NR, Theorem 2.4], the Satake isomorphism, K [ X ] W ∼ = H , and the Casselman-Shalika formula, A λ + ρ = s λ A ρ , can be formulated by the following diagram of vector space (free K -module) isomorphisms: Z ( H ) = K [ X ] W ∼ −→ H ∼ −→ ε H f f A ρ f s λ s λ A λ + ρ (2.18)This diagram has particular importance due to the fact that K [ X ] W is an avatar of theGrothendieck group of the category Rep( G ) of finite dimensional representations of G , thespherical Hecke algebra H is a form of the Grothendieck group of K -equivariant perversesheaves on the loop Grassmanian Gr , and ε H is isomorphic to the Grothendieck group ofWhittaker sheaves (appropriately formulated N -equivariant sheaves on Gr ), see [FGV].Our proof of the Casselman-Shalika formula is accomplished by restricting Theorem 1.3 tothe summand in (2.11) corresponding to − ρ ∈ A − ℓ − h . We shall identify this summand with ε H via the Z ( H )-isomorphism ε H ∼ −→ ε H p − ρ ε X µ ε X µ p − ρ Using the level ( − ℓ − h ) dot action of W from (2.9), the stabilizer of − ρ is W and W ◦ ( − ρ ) = { t − λ ◦ ( − ρ ) | λ ∈ a ∗ Z } = { ℓλ − ρ | λ ∈ a ∗ Z } . Since h ( ℓλ − ρ ) + ρ, α ∨ i ∈ ℓ Z for α ∈ R + , the straightening law (1.3) for elements of W ◦ ( − ρ ) is | s i ◦ ( ℓλ − ρ ) i = −| ℓλ − ρ i . (2.19)10teinberg product theorem in abstract Fock Space Theorem 2.3. (Casselman-Shalika) For λ ∈ ( a ∗ Z ) + and µ ∈ a ∗ Z let s λ be the Weyl character asdefined in (1.14) and let A µ be the Whittaker function as defined in (2.17) . Then s λ A ρ = A λ + ρ . Proof.
Using (2.19), | ℓλ − ρ i = ( − ℓ ( w ) ( t − ) ℓ ( w ) − ℓ ( w ) | w ◦ ( ℓλ − ρ ) i = ( − ℓ ( w ) | w ◦ ( ℓλ − ρ ) i = | ℓλ − ρ i and thus | ℓλ − ρ i satisfies the conditions of (1.7) so that C ℓλ − ρ = | ℓλ − ρ i , for λ ∈ ( a ∗ Z ) + . (2.20)By (2.14) and (2.17),[ X − ℓw µ − ρ ] = ε X µ T − w p − ρ = t − ℓ ( w ) / ε X µ p − ρ = t − ℓ ( w ) / A µ , for µ ∈ ( a ∗ Z ) + . Using (2.16), (2.20), (2.15) and that w ρ = − ρ , t − ℓ ( w ) / s λ A ρ = s λ [ X − ℓw ρ − ρ ] = s λ [ X ( ℓ − ρ ] = s λ Φ( | ( ℓ − ρ i ) = Φ( s λ | ( ℓ − ρ i )= Φ( s λ C ( ℓ − ρ ) = Φ( C − ℓw λ +( ℓ − ρ ) , by Theorem 1.3,= Φ( | ( − ℓw λ ) + ( ℓ − ρ i ) = [ X − ℓw λ +( ℓ − ρ ] = [ X − ℓw ( λ + ρ ) − ρ ]= t − ℓ ( w ) / A λ + ρ . In this section we describe the main motivation for Theorem 1.3 namely, the Steinberg-Lusztigtensor product theorem for representations of quantum groups at roots of unity. Then we explainthe connection between these results and the theory of LLT polynomials.
Let ˚ g be the Lie algebra of the group G alluded to in (1.1). Let q ∈ C × and let U q (˚ g ) be theDrinfel’d-Jimbo quantum group corresponding to ˚ g . Let∆ q ( λ ) the Weyl module for U q (˚ g ) of highest weight λ , L q ( λ ) the simple module for U q (˚ g ) of highest weight λ ,Let K (fd U q (˚ g )-mod) be the free Z [ t , t − ]-module generated by symbols [∆ q ( λ )],for λ ∈ a ∗ Z . For µ ∈ a ∗ Z , denote by W µ , resp. µ W , the set of minimal length coset representativesfor W/W µ , resp. W µ \ W . Define elements [ L q ( w y ◦ ν )], for ν ∈ A − ℓ − h and y ∈ W such that w y ∈ W ν , by the equation[∆ q ( w x ◦ ν )] = X y ≤ x X i ∈ Z ≥ " ∆ q ( w x ◦ ν ) ( i ) ∆ q ( w x ◦ ν ) ( i +1) : L q ( w y ◦ ν ) ( t ) i [ L q ( w y ◦ ν )] , M : L q ( µ )] denotes the multiplicity of the simple g -module L q ( µ ) of highest weight µ ina composition series of M and∆ q ( λ ) = ∆ q ( λ ) (0) ⊇ ∆ q ( λ ) (1) ⊇ · · · is the Jantzen filtration of ∆ q ( λ )(see, for example, [Sh, § § § §
4] for the Jantzen filtrationin this context).The combination of [LRS, (3.20)] and [LRS, Theorem 4.7]) is the following connection be-tween the representation theory of the quantum group at a root of unity and the abstract Fockspace.
Theorem 3.1.
Let ℓ ∈ Z > and let q ∈ C × such that q ℓ = 1 .Let K = Z [ t , t − ] . Then the K -linear map given by K (fd U q (˚ g )-mod) Ψ −→ F ℓ [∆ q ( λ )] λ i [ L q ( λ )] C λ is a well defined isomorphism of Z [ t , t − ] -modules. The enveloping algebra U ˚ g has a presentation by generators e , . . . , e n , f , . . . , f n and h , . . . , h n and Serre relations and the quantum group U q ˚ g has a presentation by generators E , . . . , E n , F , . . . , F n , and K , . . . , K n and quantum Serre relations such that, at q = 1, E i becomes e i and F i becomes f i . Following [Lu89] and [CP, Theorem 9.3.12], with appropriate restrictions on ℓ as in [CP, just before Proposition 9.3.5 and Theorem 9.3.12], the Frobenius map is the Hopfalgebra homomorphism
F r : U q ˚ g −→ U ˚ g E ( r ) i ( e ( r/ℓ ) i , if ℓ divides r ,0 , otherwise, F ( r ) i ( f ( r/ℓ ) i , if ℓ divides r ,0 , otherwise, K i . (3.1)The Frobenius twist of a U ˚ g -module M is the U q ˚ g -module M F r with underlying vector space M and U q ˚ g -action given by um = F r ( u ) m, for u ∈ U q ˚ g and m ∈ M . Theorem 3.2. ([Lu89, Theorem 7.4], see also [CP, 11.2.9]) Let ℓ ∈ Z > and let Π ℓ be as definedin (1.15) . Let λ ∈ ( a ∗ Z ) + and write λ = ℓλ + λ , with λ ∈ Π ℓ and λ ∈ ( a ∗ Z ) + .Let q ∈ C × be such that q ℓ = 1 and let L q ( λ ) denote the simple U q ˚ g -module of highest weight λ . Then L q ( λ ) ∼ = ∆( λ ) F r ⊗ L q ( λ ) , where ∆( µ ) denotes the irreducible U ˚ g -module of highest weight µ . Accepting Theorem 3.1, Theorem 3.2 is equivalent to the product theorem for abstract Fockspace, Theorem 1.3. 12teinberg product theorem in abstract Fock Space
In [LLT] and [LT, (43)] and [GH, Definition 6.6], the LLT polynomials for type A are defined by G ( ℓ ) µ/ν ( x, t − ) X T ∈ SSRT ℓ ( µ/ν ) t − spin( T ) x T , (3.2)where SSRT ℓ ( µ/ν ) is the set of semistandard ℓ ribbon tableaux of shape µ/ν , spin( T ) is thespin of the tableaux T and X T is the weight of the tableaux T (see [GH, § K -algebra homomorphism ψ ℓ : K [ X ] −→ K [ X ] X µ X ℓµ so that ψ ℓ ( s λ ) = char(∆ q ( λ ) F r ) , in the framework of Theorem 3.2. Then [Lcy, (57)] defines G ℓµ = X λ ∈ ( a ∗ Z ) + p ℓλ,µ s λ , (3.3)where p ℓλ,µ ∈ Z [ t ] are as in (1.7). As pointed out in [Lcy, Cor. 5.1.3], Theorem 3.1 gives ψ ℓ ( s λ ) = char(∆( λ ) F r ) = char( L q ( ℓλ )) = X µ ∈ ( a ∗ Z ) + p ℓλ,µ (1)char(∆ q ( µ )) = X µ ∈ ( a ∗ Z ) + p ℓλ,µ (1) s µ . As explained carefully in [LRS, Theorem 4.8(b)], the polynomials p ℓλ,µ are parabolic singularKazhdan-Lusztig polynomials.In [GH, Definition 5.12 and Corollary 6.4] there is another definition of LLT polynomials forgeneral Lie type: L GL,β,γ = t l β − γ + ℓ ( w ) − ℓ ( v ) X λ ∈ ( a ∗ Z ) + Q λµν s λ , where s λ ∗ · | ν i = X µ Q λµν | µ i (3.4)determine the polynomials Q λνµ . Here G is the reductive algebraic group alluded to in (1.1), L is a Levi subgroup of G with Weyl group W ν , l β − γ is the nonnegative integer defined in [GH,Remark 5.10], and µ = v ◦ ( η + ℓβ ) and ν = w ◦ ( η + kγ ) , where v ∈ W t β W η and w ∈ W t γ W η are minimal representatives. At this point, the reader’s discomfort occurring from the transitionsbetween β and γ and v and w and µ and ν is mitigated by recognizing that the relation betweenthese two definitions occurs in the special case ν = 0: Theorem 1.3 and (2.20) and the definitionof Q λµν in (3.4) give C ℓλ = s λ ∗ · C = s λ ∗ · | i = X µ Q λµ | µ i , and comparing with (1.7) gives p ℓλ,µ = Q λµ, and specifies the close relationship between G ℓµ and L GL,β,γ which occurs at ν = 0. They are thesame up to a power of t . 13teinberg product theorem in abstract Fock Space Let ˚ g be the Lie algebra of G and let g = ˚ g ⊗ C C [ ǫ, ǫ − ] + C K + C d be the corresponding affineKac-Moody Lie algebra (see [Kac, § ℓ ∈ Z > and let h be the dual Coxeter number. As explained in [LRS, Theorem 3.2], an important resultof Kazhdan-Lusztig establishes a relation between level ( − ℓ − h )-representations in paraboliccategory O g ˚ g for the affine Lie algebra and the finite dimensional representations of the quantumgroup U q ˚ g with q ℓ = 1.Let g ′ = [ g , g ] = ˚ g ⊗ C C [ ǫ, ǫ − ] + C K. By restriction, the modules in O g ˚ g are g ′ -modules. Let Λ be the fundamental weight of theaffine Lie algebra so that L ( c Λ + λ ) is an irreducible highest weight g -module of level c (i.e. K acts by the constant c ). Theorem 4.1. [KL94, Theorem 38.1] There is an equivalence of categories (cid:26) finite length g ′ -modulesof level − ℓ − h in O g ˚ g (cid:27) Ψ −→ (cid:26) finite dimensional U q (˚ g ) -moduleswith q ℓ = 1 (cid:27) ∆ g ˚ g (( − ℓ − h )Λ + λ ) ∆ q ( λ ) L (( − ℓ − h )Λ + λ ) L q ( λ )This statement of Theorem 4.1 is for the simply-laced (symmetric) case. With the propermodifications to this statement the result holds for non-simply laced cases as well, see [Lu94, § λ ∈ a ∗ Z . Under the composition of the map Ψ in Theorem 4.1 and the map Ψ fromTheorem 3.1, Ψ (Ψ ([ L (( − ℓ − h )Λ + ℓλ )])) = Ψ ([ L q ( ℓλ )]) = C ℓλ = | ℓλ i . Thus it follows from Theorem 4.1, Theorem 3.1 and (2.20) that L (( − ℓ − h )Λ + ℓλ ) = ∆ g ˚ g (( − ℓ − h )Λ + ℓλ ) = Ind g ˚ g + b ( L ˚ g ( ℓλ )) ∼ = U g ⊗ U k L ˚ g ( ℓλ ) , (4.1)where k = M k ∈ Z ≥ ǫ k (cid:16) a ⊕ M α ∈ R + ˚ g α + ˚ g − α (cid:17) with R + the set of positive roots of ˚ g .As given in (1.14), the Weyl character formula for the ˚ g -module L ˚ g ( ℓλ ) ischar( L ˚ g ( ℓλ )) = s ℓλ = Y α ∈ R + − X − α · X w ∈ W det( w ) X w ◦ ℓλ . (4.2)Letting q = e δ and using the Poincar´e-Birkhoff-Witt theorem, the character of the g -module in(4.1) ischar( L (( − ℓ − h )Λ + ℓλ )) = char(∆ g ˚ g (( − ℓ − h )Λ + ℓλ )) = char( U g ⊗ U k L ˚ g ( λ ))= s ℓλ Y k ∈ Z > − q − k ) n Y α ∈ R + − q − k X α · − q − k X − α (4.3)= Y k ∈ Z > − q − k ) n Y k ∈ Z > Y α ∈ R + − q − k X α Y k ∈ Z ≥ Y α ∈ R + − q − k X − α X w ∈ W det( w ) X w ◦ ℓλ . ⊗ and the product on right hand side is thetensor product coming from the Hopf algebra structure of U q ˚ g . Thus, in terms of affine Liealgebra representations, the Lusztig-Steinberg tensor product theorem says thatif λ ∈ ( a ∗ Z ) + and λ = λ + ℓλ with λ ∈ Π ℓ where Π ℓ is as in (1.15), then L (( − ℓ − h )Λ + λ ) ∼ = L (( − ℓ − h )Λ + λ ) ˆ ⊗ L (( − ℓ − h )Λ + ℓλ ) ∼ = L (( − ℓ − h )Λ + λ ) ˆ ⊗ ∆ g ˚ g (( − ℓ − h ) λ + ℓλ ) . (4.4) References [Bou] N. Bourbaki,
Groupes et alg`ebres de Lie , vol. 4–6, Masson 1981, MR0647314[BBF] B. Brubaker, D. Bump and S. Friedberg,
Matrix coefficients and Iwahori-Hecke al-gebra modules , Advances in Math. (2016) 247-271 doi:10.1016/j.aim.2016.05.012,arXiv:1507.07572, MR3519469[CS] W. Casselman and J. Shalika,
The unramified prinicipal series of p -adic groups II: TheWhittaker function , Compositio Math. (1980) 207-231, MR0581582.[CP] V. Chari, A. Pressley, A guide to quantum groups , Cambridge University Press, Cambridge1994, MR1358358.[FGV] E. Frenkel, D. Gaitsgory and K. Vilonen,
Whittaker patterns in the geometry of modulispaces of bundles on curves , Ann. Math. (2001), 699-748, arXiv:math.AG/9907133,MR MR1836286.[GH] I. Grojnowski and M. Haiman,
Affine Hecke algebras and positivity of LLT and Macdonaldpolynomials , 2007, available from http://math.berkeley.edu/ ∼ haiman[Gu] J. Guilhot, Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory ,arXiv:1606.05542.[HKP] T. Haines, R. Kottwitz and A. Prasad,
Iwahori-Hecke algebras , J. Ramanujan Math.Soc. (2010) 113-145, arXiv:math/0309168, MR2642451.[JM] G. James and A. Mathas, A q-analogue of the Jantzen-Schaper theorem , Proc. LondonMath. Soc. (3) (1997) 241–274, MR1425323.[Kac] V. Kac, Infinite dimensional Lie algebras , Third edition. Cambridge University Press,Cambridge, 1990. xxii+400 pp. ISBN: 0-521-37215-1; 0-521-46693-8, MR1104219[KMS] M. Kashiwara, T. Miwa and E. Stern,
Decomposition of q -deformed Fock spaces ,arxiv:q-alg/9508006, Selecta Math. (1996) 787–805, MR1383585.[KT95] M. Kashiwara, T. Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras withnegative level , Duke Math. J. (1995) 21–62, MR1317626.15teinberg product theorem in abstract Fock Space[KL94] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras I, II, IIIand IV , J. Amer. Math. Soc. (1993) 905–947 and 949–1011 MR1186962, J. Amer. Math.Soc. (1994) 335–381 MR1239506, and J. Amer. Math. Soc. (1994) 383–453, MR1239507[Knp] F. Knop, On the Kazhdan-Lusztig basis of a spherical Hecke algebra
Represent. Theory (2005) 417?425, arXiv:math/0403066, MR2142817.[LRS] M. Lanini, A. Ram and P. Sobaje, Fock space model for decomposition numbers for quan-tum groups at roots of unity , to appear in Kyoto Math. J., arXiv:1612.03120.[LLT] A. Lascoux, B. Leclerc, J-Y. Thibon,
Ribbon tableaux, Hall-Littlewood functions,quantum affine algebras and unipotent varieties , J. Math. Phys. (1997) 1041-1068,arXiv:q-alg/9512031, MR1399754.[LT] B. Leclerc, J-Y. Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polyno-mials , Combinatorial methods in representation theory (Kyoto, 1998), 155–220, Adv. Stud.Pure Math., , Kinokuniya, Tokyo, 2000, arXiv:math/9809122, MR1864481.[Lcy] C. Lecouvey, Parabolic Kazhdan-Lusztig polynomials, plethysms and generalised Hall-Littlewood functions for classical types , European J. Combinatorics (2009) 157-191,arXiv:math.RT/0607038, MR2460224.[Li] P. Littelmann, Paths and root operators in representation theory , Ann. Math. (1995)499-525, MR1356780.[Lu89] G. Lusztig,
Modular representations and quantum groups , Contemp. Math. (1989)58–77, MR0982278.[Lu94] G. Lusztig, Monodromic systems on affine flag manifolds , Proc. R. Soc. Lond. A (1994) 231–246, MR1276910.[Lu95] G. Lusztig,
Errata: “Monodromic systems on affine flag manifolds” [Proc. Roy. Soc.London Ser. A 445 (1994), no. 1923, 231–246; MR1276910] , Proc. Roy. Soc. London Ser.A (1995) 731–732, MR2105507.[NR] K. Nelsen and A. Ram,
Kostka-Foulkes polynomials and Macdonald spherical functions ,Surveys in combinatorics, 2003 (Bangor), 325–370, London Math. Soc. Lecture Note Ser., , Cambridge Univ. Press, Cambridge 2003, arXiv:0401298, MR2011741.[NP] B.C. Ngo and P. Polo,
R´esolutions de Demazure affines et formule de Casselman-Shalikag´eom´etrique , J. Algebraic Geometry (2001) 515-547, math.AG/0005022, MR1832331.[Ra] A. Ram, Alcove walks, Hecke algebras, Spherical functions, crystals and column stricttableaux , Pure and Applied Mathematics Quarterly (Special Issue: In honor of RobertMacPherson, Part 2 of 3) no. 4 (2006) 963-1013, arXiv:0601.343, MR2282411.[Sh] P. Shan, Graded decomposition matrices of v -Schur algebras via Jantzen filtration , Repre-sentation Theory16