STRUCTURE CONSTANTS FOR HECKE AND REPRESENTATION RINGS

THOMAS J. HAINES

Abstract. In [KLM] the authors study certain structure constants for two related rings: the spherical Hecke algebra of a split connected reductive group over a local non-Archimedean field, and the representation ring of the Langlands dual group. The former are defined relative to characteristic functions of double cosets, and the latter relative to highest weight representations. They prove that the nonvanishing of one of the latter structure constants always implies the nonvanishing of the corresponding former one. For GL_{n}, the reverse implication also holds, and is due to P. Hall. Both proofs are combinatorial in nature. In this note, we provide geometric proofs of both results, using affine Grassmannians. We also provide some additional results concerning minuscule coweights and the equidimensionality of the fibers of certain Bott-Samelson resolutions of affine Schubert varieties for GL_{n}.

1. Introduction

Given a split connected reductive group G over a local non-Archimedean field F, one may consider two related rings: the spherical Hecke algebra attached to G(F), and the representation ring of the Langlands dual group Ĝ. By the Satake isomorphism, these rings are actually isomorphic. However, the two realizations give rise naturally to two different bases and thus to two different sets of structure constants defining this ring. For the Hecke algebra realization, consider the characteristic functions of double cosets with respect to a special maximal compact subgroup; for the representation ring realization, consider the highest weight representations. In [KLM], the authors prove several new facts concerning these structure constants and their relation to each other, culminating in a new proof of the saturation property for the group GL_{n}. They prove that the non-vanishing of a representation ring structure constant implies the non-vanishing of the corresponding Hecke algebra structure constant. For GL_{n}, the converse implication also holds, and is due to P. Hall. Both proofs are combinatiorial in nature.

The purpose of this paper is to provide geometric proofs for both results, using affine Grassmannians. Along the way, we provide some additional results concerning minuscule coweights and the equidimensionality of the fibers of certain Bott-Samelson resolutions of affine Schubert varieties for GL_{n}. These geometric methods avoid the combinatorial tools which are specific to the group GL_{n}, and so give some indications how one may approach proving saturation properties for more general groups.

More precisely, let G denote a connected reductive group over an algebraic closure k of a finite field 𝔽_{q}. We assume G is defined and split over 𝔽_{q}. Let T denote an 𝔽_{q}-split maximal torus of G. Let B = T U denote an 𝔽_{q}-rational Borel subgroup containing T. Let X_{∗}(T) denote the set of cocharacters of T, and let X_{+}^{∨} denote the set of Bdominant elements of X_{∗}(T). Let ρ denote half the sum of the B-positive roots for G. Let ⟨·, ·⟩ : X^{∗}(T) × X_{∗}(T) → ℤ denote the canonical pairing.

Fix a prime ℓ different from the characteristic of k, and let Ĝ = Ĝ(ℚ̄_{ℓ}) denote the Langlands dual of G over the field ℚ̄_{ℓ} (an algebraic closure of ℚ_{ℓ}). Each dominant cocharacter μ ∈ X_{+}^{∨} can be thought of as a dominant weight for Ĝ, hence such a cocharacter gives rise to a unique irreducible ℚ̄_{ℓ}-linear representation V_{μ} of Ĝ having highest weight μ. Let Rep(Ĝ) denote the category of rational representations of Ĝ over ℚ̄_{ℓ}.

We will work with the spherical Hecke algebra of G in the function-field setting. Let O = k[[t]] and F = k((t)). Also, let O_{q} = 𝔽_{q}[[t]] and F_{q} = 𝔽_{q}((t)). Let K := G(O) denote the “maximal compact” subgroup of G(F). Let K_{q} = G(O_{q}). We let H_{q} denote the Hecke algebra for G over F_{q}, i.e., the convolution algebra of compactly-supported K_{q}-bi-invariant ℚ̄_{ℓ}-valued functions on G(F_{q}) (convolution is defined using the Haar measure on G(F_{q}) which assigns measure one to the compact open subgroup K_{q}). For μ ∈ X_{+}^{∨}, define an element f_{μ} ∈ H_{q} by

f_{μ} = char(K_{q} t_{μ} K_{q})

where t_{μ} := μ(t) ∈ T (F_{q}). By the Cartan decomposition, the elements f_{μ} form a ℚ̄_{ℓ}- basis for H_{q}. For later use let us recall that the convolution operation ∗ is defined, for f_{1}, f_{2} ∈ H_{q}, by

(f_{1} ∗ f_{2})(g) = ∫_{G(F_{q})} f_{1}(x)f_{2}(x^{−1}g)dx.

Now let μ_{1}, …, μ_{r} and λ be dominant coweights for G. Denote μ_{•} = (μ_{1}, …, μ_{r}) and |μ_{•}| = μ_{1} + ⋯ + μ_{r}. We consider the decomposition in Rep(Ĝ)

V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}} = ⨁_{λ∈X_{+}^{∨}, λ≤|μ_{•}|} V_{λ} ⊗ V_{μ_{•}}^{λ},

where V_{μ_{•}}^{λ} denotes the multiplicity vector space. Here λ_{1} ≤ λ_{2} means that λ_{2} − λ_{1} is a sum of B-positive coroots. (It follows from the discussion in section 2 that if V_{λ} occurs as a summand, then λ ≤ |μ_{•}| necessarily holds.) The numbers dim(V_{μ_{•}}^{λ}) are the structure constants for the representation ring of Ĝ.

In a parallel manner, for μ_{•} and λ as above, we write

f_{μ_{1}} ∗ ⋯ ∗ f_{μ_{r}} = ∑_{λ} c_{μ_{•}}^{λ}f_{λ},

for certain constants c_{μ_{•}}^{λ}; in this paper these are referred to as the structure constants for the Hecke algebra H_{q}. It is known and easy to see that the structure constants all belong to ℤ. In fact, as q varies, they are given by polynomial functions in q with integer coefficients: c_{μ_{•}}^{λ} = c_{μ_{•}}^{λ}(q). (See Lemmas 9.15, 9.18 of [KLM]).

The purpose of this paper is to study the relation between the following two properties of a collection (μ_{•}, λ) of dominant cocharacters:

Rep(μ_{•}, λ): The irreducible representation V_{λ} occurs with non-zero multiplicity in V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}}. That is, dim(V_{μ_{•}}^{λ}) > 0.

Hecke(μ_{•}, λ): The function f_{λ} appears with non-zero coefficient in f_{μ_{1}} ∗ ⋯ ∗ f_{μ_{r}}. That is, c_{μ_{•}}^{λ} ≠ 0. Equivalently, K_{q} t_{λ} K_{q} ⊂ K_{q} t_{μ_{1}} K_{q} ⋯ K_{q} t_{μ_{r}} K_{q}.

The following result was first proved for general groups G by M. Kapovich, B. Leeb, and J. Millson in [KLM], Theorem 9.19. In this paper we give another approach, based on the geometry of affine Grassmannians.

Theorem 1.1 (KLM). If Rep(μ_{•}, λ) holds, then Hecke(μ_{•}, λ) also holds. More precisely, we have the following description of the Hecke algebra structure constants

c_{μ_{•}}^{λ}(q) = dim(V_{μ_{•}}^{λ}) q^{⟨ρ,|μ_{•}|−λ⟩} + {terms with lower q-degree}.

For the general linear group, the converse implication also holds, and is originally due to P. Hall, using the combinatorics of Hall polynomials (cf. [KLM], section 9.6, and [Mac], part II, Theorem (4.3)). We present a geometric proof here, again using affine Grassmannians.

Theorem 1.2 (P. Hall). Let G = GL_{n}. Then Hecke(μ_{•}, λ) ⇒ Rep(μ_{•}, λ).

The approach of this paper is to reformulate the two properties Hecke(μ_{•}, λ) and Rep(μ_{•}, λ) in terms of properties of the “multiplication” morphism

m_{μ_{•}} : Q̃_{μ_{•}} → Q̄_{|μ_{•}|}

used to define the convolution of K-equivariant perverse sheaves on the affine Grassmannian. Here the domain is a twisted product of the closures Q̄_{μ_{i}} of K-orbits Q_{μ_{i}} in the affine Grassmannian for G. The morphism m_{μ_{•}} is given by forgetting all but the last factor in the twisted product. It is a (stratified) semi-small, proper, and birational morphism (cf. section 2). One can view it as a partial desingularization of Q̄_{|μ_{•}|}, directly analogous to the Bott-Samelson partial desingularizations of Schubert varieties. (When the coweights μ_{i} are all minuscule, the domain is smooth, so in that case m_{μ_{•}} is a genuine resolution of singularities.) The semi-smallness of m_{μ_{•}} means that the fiber over any point y in a K-orbit stratum Q_{λ} ⊂ Q̄_{|μ_{•}|} has dimension bounded above by ⟨ρ, |μ_{•}| − λ⟩.

The two properties can be translated into properties of the fibers of m_{μ_{•}}. The geometric Satake isomorphism (cf. [Gi], [MV], [NP]) plays a key role in this step.

Proposition 1.3. Suppose (μ_{•}, λ) is such that Q_{λ} ⊂ Q̄_{|μ_{•}|} (equivalently, λ ≤ |μ_{•}|). Fix y ∈ Q_{λ}. Then:

1. V_{μ_{•}}^{λ} ≠ 0 if and only if m_{μ_{•}}^{−1}(y) has an irreducible component of dimension ⟨ρ, |μ_{•}|− λ⟩. Moreover, such a component necessarily meets the open stratum Q_{μ_{•}}.

2. Suppose y is 𝔽_{q}-rational. Then f_{λ} occurs in f_{μ_{1}} ∗ ⋯ ∗ f_{μ_{n}} if and only if m_{μ_{•}}^{−1}(y) meets the open stratum Q_{μ_{•}}. In that case, we have

c_{μ_{•}}^{λ}(q) = #(m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}})(𝔽_{q}) > 0.

The first part of Theorem 1.1 follows immediately from Proposition 1.3. We deduce the second part concerning c_{μ_{•}}^{λ} with little additional effort, by using the Weil conjectures to approximate the number of points on the algebraic varieties m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}}.

We also have the following partial converse to Theorem 1.1, valid for every group G. It is an easy and purely combinatorial consequence of the P-R-V conjecture (now a theorem due independently to S. Kumar and O. Mathieu, cf. [Lit]).

Proposition 1.4. If Hecke(μ_{•}, λ) holds, then there exist dominant coweights μ′_{i} ≤ μ_{i} (1 ≤ i ≤ r), such that Rep(μ′_{•}, λ) holds.

To prove Theorem 1.2, we need finer information about the fibers of m_{μ_{•}} in the GL_{n} case. The first ingredient is the following proposition valid for all groups having minuscule coweights (recall that a coweight μ is minuscule provided that ⟨α, μ⟩ ∈ {−1, 0, 1}, for every root α of G). It is also a consequence of the P-R-V conjecture.

Proposition 1.5. If μ_{1}, …, μ_{r} are all minuscule coweights for G, and λ ≤ |μ_{•}|, then both Rep(μ_{•}, λ) and Hecke(μ_{•}, λ) hold.

In view of Proposition 1.3, this implies:

Corollary 1.6. Let μ_{1}, …, μ_{r} let be minuscule coweights for G, and let λ ≤ |μ_{•}|. Then for any y ∈ Q_{λ}, we have

dim(m_{μ_{•}}^{−1}(y)) = ⟨ρ, |μ_{•}| − λ⟩.

In particular, every stratum of Q̄_{|μ_{•}|} is relevant for the semi-small morphism m_{μ_{•}}.

We make essential use of the further information that in the GL_{n} case, the fibers above are equidimensional.

Proposition 1.7. Let G = GL_{n}. Let μ_{1}, …, μ_{r} be minuscule coweights. Suppose Q_{λ} ⊂ Q̄_{|μ_{•}|}, and let y ∈ Q_{λ}. Then every irreducible component of the fiber

m_{μ_{•}}^{−1}(y)

has dimension ⟨ρ, |μ_{•}| − λ⟩.

The proof of Proposition 1.7 proceeds by reduction to a theorem of N. Spaltenstein [Sp] concerning partial Springer resolutions of the nilpotent cone for GL_{n}. The lack of an analogous result for other groups is one reason Proposition 1.7 can be proved (at the moment) only for GL_{n}.

In fact for the general linear group, Proposition 1.7 can be used to prove a seemingly stronger result.

Proposition 1.8. Let μ_{1}, …, μ_{r} be dominant coweights for G = GL_{n}. Let y ∈ Q_{λ} ⊂ Q̄_{|μ_{•}|}. Let Q_{μ′_{•}} ⊂ Q̃_{μ_{•}} be the stratum indexed by μ′_{•} = (μ′_{1}, …, μ′_{r}) for dominant coweights μ′_{i} ≤ μ_{i} (1 ≤ i ≤ r). Then any irreducible component of the fiber m_{μ_{•}}^{−1}(y) whose generic point belongs to Q_{μ′_{•}} has dimension ⟨ρ, |μ′_{•}| − λ⟩.

As we explain in section 8, Proposition 1.8 quickly implies Theorem 1.2.

Proposition 1.8 appears to be special to GL_{n}. Indeed, it is actually somewhat stronger than the implication Hecke(μ_{•}, λ) ⇒ Rep(μ_{•}, λ), which is known to fail in general (e.g. for SO(5) or G_{2}, cf. [KLM], section 9.5). But Proposition 1.7 could remain valid if GL_{n} is replaced with an arbitrary group G, and it would be interesting to clarify the situation. Such a generalization of Proposition 1.7 would have applications to proving a type of “Saturation theorem” for a general reductive group (along the same lines as the new proof in [KLM] of the Saturation theorem for GL_{n}, originally proved by Knutson-Tao [KT]).

Remark. Suitably reformulated, the main results of this paper hold if the coefficient field k = 𝔽̄_{q} is replaced by any algebraically closed field κ (e.g. ℂ). We let now O = κ[[t]], K = G(O), etc., and replace Hecke(μ_{•}, λ) with

Hecke′(μ_{•}, λ) : Kt_{λ}K ⊂ Kt_{μ_{1}}K ⋯ Kt_{μ_{r}}K. Then the arguments of this paper prove that Rep(μ_{•}, λ) ⇒ Hecke′(μ_{•}, λ) for every G, and that Rep(μ_{•}, λ) ⇔ Hecke′(μ_{•}, λ) for GL_{n}. The statements concerning dimensions of fibers and their equidimensionality also remain valid. Thus, here we avoid the hypothesis that F_{q} have finite residue field, essential for Hall’s proof of Theorem 1.2 using Hall polynomials, and for the proof of Theorem 1.1 in ([KLM], Theorem 9.19).

Acknowledgments. It is a pleasure to thank John Millson for several stimulating conversations, and for delivering a series of lectures on [KLM] at the University of Maryland which sparked my interest in this subject. My intellectual debt to B.C. Ngô and P. Polo will be clear to the reader familiar with [Ngo] or [NP], where several of the ideas in this paper appear, at least implicitly. I also thank J.K. Yu for some helpful conversations, and M. Rapoport for his comments on the paper. Finally, I am grateful to R. Kottwitz for suggesting some significant improvements in exposition, and for making detailed comments on the first version of this paper.

2. Review of affine Grassmannians

In this section we recall some well-known notions relating to affine Grassmannians. The reader can find further details in [Gi], [MV], and [NP].

We will work with the affine Grassmannian

Q = G(F)/K,

which can be thought of as the k-points of an ind-scheme defined over 𝔽_{q}. The group scheme K = G(O) acts naturally on Q (on the left). By the Cartan decomposition, the K-orbits are parametrized by dominant coweights λ ∈ X_{+}^{∨}. Indeed, let e_{0} denote the base point of Q corresponding to the coset K, and let e_{λ} = t_{λ}e_{0}. Then Q_{λ} = Ke_{λ} is the K-orbit corresponding to λ. It is well-known that Q_{λ} is a smooth quasi-projective variety of dimension ⟨2ρ, λ⟩, defined over 𝔽_{q}. Its closure Q̄_{λ} in Q is projective, but in general is not smooth.

Let j_{λ} : Q_{λ} ↪ Q̄_{λ} denote the open immersion into the closure, and let us define the intersection complex

A_{λ} := IC_{λ}(ℚ̄_{ℓ}) = j_{λ,!∗}(ℚ̄_{ℓ})[⟨2ρ, λ⟩].

Here we are applying j_{λ,!∗}, the Goresky-MacPherson middle extension functor (for the middle perversity, cf. [GM], [BBD]), to the shifted constant sheaf on the smooth variety Q_{λ}. The cohomological shift by the dimension of Q_{λ} is to ensure the result is a perverse sheaf. It is known that A_{λ} is a self-dual K-equivariant simple perverse sheaf on Q.

Let P_{K}(Q) denote the category of ℚ̄_{ℓ}-linear K-equivariant perverse sheaves on Q. This is a semi-simple abelian category (cf. [Ga]), whose simple objects are precisely the intersection complexes A_{λ}, for λ ∈ X_{+}^{∨}. In fact there exists a tensor (or “convolution”) operation

⋆ : P_{K}(Q) × P_{K}(Q) → P_{K}(Q)

(defined below) which gives P_{K}(Q) the structure of a neutral Tannakian category over ℚ̄_{ℓ}. The following theorem which identifies the corresponding algebraic group as the Langlands dual plays a key role in this paper. We refer the reader to [Gi],[MV], [NP], and the appendix of [Nad], for details of the proof.

Theorem 2.1 (Geometric Satake Isomorphism). There is an equivalence of tensor categories

(P_{K}(Q), ⋆) ˜⟶ (Rep(Ĝ), ⊗),

under which A_{λ} corresponds to the irreducible representation V_{λ} with highest weight λ.

2.1. Definition of the twisted product. To define the operation ⋆, we need some more preliminaries. First, recall that any ordered pair of elements L, L′ ∈ Q gives rise to an element inv(L, L′) ∈ X_{+}^{∨} (the “relative position” of L, L′). The map inv : Q × Q → X_{+}^{∨} is defined as the composition

inv : G(F)/K × G(F)/K → G(F)\[G(F)/K × G(F)/K] = K\G(F)/K ˜→X_{+}^{∨},

where G(F) acts diagonally on G(F)/K × G(F)/K. For G = GL_{n}, inv(L, L′) is just the usual relative position of two O-lattices in F^{n} given by the theory of elementary divisors.

The usual partial ordering ≤ on the set of dominant coweights corresponds to the closure relation in Q: Q_{λ} ⊂ Q̄_{μ} if and only if λ ≤ μ. It follows that inv(L, L′) ≤ μ if and only if there exists g ∈ G(F) such that gL = e_{0} and gL′ ∈ Q̄_{μ}. Thus, for L ∈ Q and μ ∈ X_{+}^{∨} fixed, the set of L′ with inv(L, L′) ≤ μ can be thought of as the k-points of a projective algebraic variety, isomorphic to Q̄_{μ}.

Now let μ_{•} = (μ_{1}, …, μ_{r}), where μ_{i} ∈ X_{+}^{∨} for 1 ≤ i ≤ r. We define the twisted product scheme

Q̃_{μ_{•}} = Q̄_{μ_{1}} ˜× ⋯ ˜×Q̄_{μ_{n}}

to be the subscheme of Q^{n} consisting of points (L_{1}, …, L_{r}) such that inv(L_{i−1}, L_{i}) ≤ μ_{i} for 1 ≤ i ≤ r (letting L_{0} = e_{0}). Projection onto the last coordinate gives a proper surjective birational map

m_{μ_{•}} : Q̃_{μ_{•}} → Q̄_{|μ_{•}|}.

The birationality follows from the more precise statement that the restriction of m_{μ_{•}} to the inverse image of the open stratum Q_{|μ_{•}|} ⊂ Q̄_{|μ_{•}|} is an isomorphism. This in turn can be deduced from the corresponding statement for an analogous Demazure resolution of an affine Schubert variety in the affine flag variety. We omit the details.

2.2. Semi-small morphisms. We shall make use of the notion of semi-small morphisms. Let f : X = ∪_{α}X_{α} → Y = ∪_{β}Y_{β} be a proper surjective birational morphism between stratified spaces. Suppose each f(X_{α}) is a union of strata Y_{β}. We say f is semi-small if, whenever y ∈ Y_{β} ⊂ f(X_{α}), then

dim(f^{−1}(y) ∩ X_{α}) ≤ \frac{1}{2}(dim(X_{α}) − dim(Y_{β})).

(In [MV], this notion is termed “stratified semi-small”. In the usual terminology ([GM]), the domain of a semi-small morphism is assumed to be smooth.)

We say f is locally trivial (in the stratified sense) if whenever Y_{β} ⊂ f(X_{α}), the restriction of f : f^{−1}(Y_{β}) → Y_{β} to X_{α} ∩ f^{−1}(Y_{β}) is Zariski-locally a trivial fibration. In this case we have, for every y ∈ Y_{β} ⊂ f(X_{α}):

dim(f^{−1}(y) ∩ X_{α}) + dim(Y_{β}) = dim(f^{−1}(Y_{β}) ∩ X_{α}).

The target Q̄_{|μ_{•}|} of m_{μ_{•}} is stratified by the locally-closed subschemes Q_{λ} (λ ≤ |μ_{•}|), and the domain Q̄_{μ_{1}} ˜× ⋯ ˜×Q̄_{μ_{n}} is stratified by the locally-closed subspaces Q_{μ′_{•}} = Q_{μ′_{1}} ˜× ⋯ ˜×Q_{μ′_{n}}, where μ′_{•} = (μ′_{1}, …, μ′_{n}) satisfies μ′_{i} ≤ μ_{i}, for every i. (The definition of the subspace is the same as that of the ambient space, except that the inequalities inv(L_{i−1}, L_{i}) ≤ μ_{i} are replaced by the equalities inv(L_{i−1}, L_{i}) = μ′_{i}.) With respect to these stratifications, we have the following result.

Proposition 2.2 ([MV], [NP]). The morphism m_{μ_{•}} is a semi-small and locally trivial morphism.

Remark. The proof of semi-smallness over 𝔽_{q} does not appear explicitly in the literature in precisely this generality. The proof of Lemme 9.3 in [NP] for the special case where all μ_{i} are minuscule or quasi-minuscule works as well for the general case, modulo the inequality dim(S_{ν} ∩ Q_{λ}) ≤ ⟨ρ, λ + ν⟩, where ν ∈ X_{∗}(T) and S_{ν} := U(F)e_{λ}. In fact the equality dim(S_{ν} ∩Q_{λ}) = ⟨ρ, λ+ν⟩ can be deduced (as Ngô and Polo remark), from their Theoreme 3.1. This indirect method to prove the dimension equality is not circular, since Ngô and Polo use the equality dim(S_{ν} ∩ Q_{λ}) = ⟨ρ, λ + ν⟩ only in the special case mentioned above in their proof of Theoreme 3.1, and they prove this special case by direct means.

2.3. Convolution of perverse sheaves. If G_{i} (1 ≤ i ≤ r) are elements of P_{K}(Q), choose coweights μ_{i} such that supp(G_{i}) ⊂ Q̄_{μ_{i}}, for every i (technical aside: this is possible, as we can assume with no loss of generality that each G_{i} is supported on only one connected component of Q). There is a unique “twisted product” perverse sheaf G_{1} ˜⊠ ⋯ ˜⊠G_{r} on the twisted product Q̃_{μ_{•}}, which is locally isomorphic to the usual exterior product G_{1} ⊠ ⋯ ⊠ G_{r} on the product space Q̄_{μ_{1}} × ⋯ × Q̄_{μ_{r}}. (See 7.4 of [HN], or [NP], section 2, for another construction). We then define

G_{1} ⋆ ⋯ ⋆ G_{r} = Rm_{μ_{•},∗}(G_{1} ˜⊠ ⋯ ˜⊠G_{r}).

This belongs to P_{K}(Q) by the semi-smallness of m_{μ_{•}}, and is independent of the choice of μ_{•}.

The important lemma below follows directly from the definitions and the following characterization of the intersection complex ([BBD], 2.1.17): Let X^{0} denote the open (smooth) stratum in a stratified variety X = ∪_{α}X_{α}. Then IC(X) is the unique self-dual perverse extension of ℚ̄_{ℓ}[dim(X)] on X^{0} which satisfies, for each stratum X_{α} ≠ X^{0}, the property H^{i}IC(X)|_{X_{α}} = 0 for i ≥ −dim(X_{α}).

Lemma 2.3. A_{μ_{1}} ⋆ ⋯ ⋆ A_{μ_{r}} = Rm_{μ_{•},∗}IC(Q̃_{μ_{•}}).

We are going to use this to give a geometric description of the multiplicity space V_{μ_{•}}^{λ}, which will be the key step to proving Proposition 1.3.

3. Proof of Proposition 1.3, Part (1)

The key result is the following well-known proposition. It has appeared without proof in several published articles (e.g. [NP]). We provide a proof here for the convenience of the reader.

Proposition 3.1. Fix λ ≤ |μ_{•}|, and y ∈ Q_{λ}. Then the space V_{μ_{•}}^{λ} has a basis in canonical bijection with the set of irreducible components of m_{μ_{•}}^{−1}(y) having maximal possible dimension, that is, ⟨ρ, |μ_{•}| − λ⟩.

Let us assume Proposition 3.1 and deduce Proposition 1.3, Part (1). The first statement is obvious, so we verify the second. But it is clear that an irreducible component C of m_{μ_{•}}^{−1}(y) with dimension ⟨ρ, |μ_{•}| − λ⟩ meets the open stratum Q_{μ_{•}} (if not, then C meets only strictly smaller strata Q_{μ′_{•}}, and then by Proposition 2.2, the dimension of C would be bounded above by

dim(m_{μ_{•}}^{−1}(y) ∩ Q_{μ′_{•}}) ≤ ⟨ρ, |μ′_{•}| − λ⟩ < ⟨ρ, |μ_{•}| − λ⟩.)

Proof of Proposition 3.1: By Theorem 2.1 we have

A_{μ_{1}} ⋆ ⋯ ⋆ A_{μ_{r}} = ⨁_{λ≤|μ_{•}|} A_{λ} ⊗ V_{μ_{•}}^{λ}.

Let d = −dim(Q_{λ}). Now apply H_{y}^{d} (take the dth cohomology stalk at y) to both sides of this equation.

The right hand side gives the vector space V_{μ_{•}}^{λ}, since A_{λ} restricted to Q_{λ} is ℚ̄_{ℓ}[dim(Q_{λ})] (the constant sheaf placed in degree −dim(Q_{λ})) and since by definition of intersection complexes, H_{y}^{d}A_{λ′} is zero if λ < λ′. (In general, H^{i}IC(X)|_{X_{α}} = 0 if i > −dim(X_{α}) and also for i ≥ −dim(X_{α}) if X_{α} is not the open stratum in X.)

By Lemma 2.3, on the left hand side we get

H^{d}(m_{μ_{•}}^{−1}(y), IC(Q̃_{μ_{•}})).

The result now follows by virtue of the following lemma.

Lemma 3.2. Let f : X = ∪_{α}X_{α} → Y = ∪_{β}Y_{β} be a semi-small morphism between proper stratified schemes, and suppose y ∈ Y_{β}. Let d = −dim(Y_{β}). Then there is a canonical isomorphism

H^{d}(f^{−1}(y), IC(X)) = ℚ̄_{ℓ}^{C_{max}(y)},

where C_{max}(y) is the set of irreducible components of f^{−1}(y) having the maximal possible dimension \frac{1}{2}(dim(X) − dim(Y_{β})).

Proof. Let X^{0} denote the open (smooth) stratum of X; recall that IC(X)|_{X^{0}} = ℚ̄_{ℓ}[dim(X)]. Suppose we fix X_{α} ≠ X^{0}. We claim that

H^{d}(f^{−1}(y) ∩ X_{α}, IC(X)) = 0.

Indeed, if not, then the “local-global” spectral sequence

H^{p}(f^{−1}(y) ∩ X_{α}, H^{q}IC(X)) ⇒ H^{p+q}(f^{−1}(y) ∩ X_{α}, IC(X))

shows that there exists q and p = d − q such that the initial term is non-zero. We have therefore q < −dim(X_{α}), which together with the semi-smallness of f gives

p ≤ 2(dim(f^{−1}(y) ∩ X_{α})) ≤ dim(X_{α}) − dim(Y_{β}) < −q − dim(Y_{β}),

which is clearly impossible since p + q = d = −dim(Y_{β}).

The same argument shows that H^{d}(f^{−1}(y) ∩ (X\X^{0}), IC(X)) = 0.

Since f^{−1}(y) and f^{−1}(y) ∩ (X\X^{0}) are proper, their cohomology and compactlysupported cohomology agree. It follows from the above remarks that

H^{d}(f^{−1}(y), IC(X)) = H_{c}^{d}(f^{−1}(y) ∩ X^{0}, ℚ̄_{ℓ}[dim(X)]) = H_{c}^{d+dim(X)}(f^{−1}(y) ∩ X^{0}, ℚ̄_{ℓ}),

from which the lemma follows easily. □

4. Proof of Proposition 1.3, Part (2)

The next lemma follows from the definition of convolution in H_{q}, and is left to the reader.

Lemma 4.1. c_{μ_{•}}^{λ}(q) = #(m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}})(𝔽_{q}).

Therefore, if f_{λ} occurs in f_{μ_{1}} ∗ ⋯ ∗ f_{μ_{r}}, we obviously have m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}} ≠ ∅.

Conversely, suppose that m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}} ≠ ∅. It follows from this that c_{μ_{•}}^{λ}(q^{n}) > 0 for n >> 0. Using the definition of convolution, this means that

t_{λ} ∈ K_{q^{n}}t_{μ_{1}}K_{q^{n}} ⋯ K_{q^{n}}t_{μ_{r}}K_{q^{n}}.

But by the discussion below, this condition can be expressed purely in terms of the extended affine Weyl group, and in particular, it is independent of n. Therefore, it holds for n = 1 and thus c_{μ_{•}}^{λ}(q) > 0.

To complete the argument proving independence of n, let W̃ = X_{∗}(T) ⋊ W denote the extended affine Weyl group. Denote the translation element in W̃ corresponding to μ ∈ X_{∗}(T) also by the symbol t_{μ}. Let I_{q^{n}} ⊂ K_{q^{n}} be the Iwahori subgroup defined to be the inverse image of B under the homomorphism G(𝔽_{q^{n}}[[t]]) → G(𝔽_{q^{n}}) given by t ↦ 0. For simplicity, write K = K_{q^{n}} and I = I_{q^{n}}.

Then K = IW I := ∐_{w∈W} IwI and G(𝔽_{q^{n}}((t))) = IW̃I := ∐_{w∈W̃} IwI. Furthermore, standard results for BN pairs yield the identity

IxIyI ⊂ ∐_{ỹ⪯y} IxỹI,

for x, y ∈ W̃, where ỹ ranges over elements preceding y in the Bruhat order ^{1} on W̃. Using this we see

t_{λ} ∈ Kt_{μ_{1}}K ⋯ Kt_{μ_{r}}K ⇔ t_{λ} ∈ IW t_{μ_{1}}W I ⋯ IW t_{μ_{r}}W I
⇔ ∃ x_{i} ∈ W t_{μ_{i}}W such that t_{λ} ∈ Ix_{1}I ⋯ Ix_{r}I.

But by standard facts for BN-pairs, the set S ⊂ W̃ appearing in the union

Ix_{1}I ⋯ Ix_{r}I = ∐_{w∈S} IwI

depends only on the elements x_{1}, …, x_{r} ∈ W̃ (and not on the power q^{n} in I = I_{q^{n}}).

5. End of proof of Theorem 1.1

It remains to prove the formula

c_{μ_{•}}^{λ}(q) = dim(V_{μ_{•}}^{λ}) q^{⟨ρ,|μ_{•}|−λ⟩} + {terms of lower q-degree}.

Clearly we may prove this after base extension (enlarging q), so that we may assume the irreducible components of m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}} are defined over 𝔽_{q}. But then taking Lemma 4.1 and Proposition 3.1 into account, the formula follows immediately from the following lemma, itself a consequence of Deligne’s paper [Weil2].

Lemma 5.1. If X is a geometrically irreducible 𝔽_{q}-variety, then the function #X(𝔽_{q}) is of the form q^{dim(X)} + r(q), where |r(q)| ≤ O(q^{dim(X)−1/2}).

6. The P-R-V conjecture and Propositions 1.4 and 1.5

The Parthasarathy-Ranga-Rao-Varadarajan (P-R-V) conjecture has been proved independently by S. Kumar and O. Mathieu. See [Lit], section 10, for a short proof using the Littelmann path model.

1The Bruhat order here is defined using the set of simple affine reflections through the hyperplanes bounding the opposite base alcove, i.e., the alcove w_{0}(A), where A is the alcove fixed by I.

Theorem 6.1 (P-R-V Conjecture). For 1 ≤ i ≤ r, let μ_{i} be a dominant coweight, and let w_{i} ∈ W be an element of the finite Weyl group. Suppose ν = w_{1}μ_{1} + ⋯ + w_{r}μ_{r} is dominant. Then V_{ν} occurs with multiplicity at least one in the tensor product

V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}}.

(This is usually stated only in the case r = 2, but the above version follows easily by induction on r.)

Theorem 6.1 is the main ingredient to proving Proposition 1.4 and the assertion Rep(μ_{•}, λ) in Proposition 1.5. Indeed, for the latter we only need to verify the following lemma to be able to take ν = λ in Theorem 6.1.

Lemma 6.2. Suppose that each dominant coweight μ_{i} is minuscule, and that λ is dominant and satisfies λ ≤ μ_{1}+⋯+μ_{r}. Then there exist elements w_{i} ∈ W (1 ≤ i ≤ r), such that

λ = w_{1}μ_{1} + ⋯ + w_{r}μ_{r}.

For V ∈ Rep(Ĝ), let Ω(V) ⊂ X^{∗}(T̂) = X_{∗}(T) denote the set of its weights with respect to the dual torus T̂. Recall that

Ω(V_{μ}) = {ν ∈ X^{∗}(T̂) | wν ≤ μ, ∀w ∈ W }.

The following well-known lemma is left to the reader.

Lemma 6.3. Let μ_{1}, …, μ_{r} be dominant coweights (also viewed as weights for Ĝ). Then

Ω(V_{|μ_{•}|}) = Ω(V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}}) = {ν_{1} + ⋯ + ν_{r} | ν_{i} ∈ Ω(V_{μ_{i}}), 1 ≤ i ≤ r}.

Note that any dominant λ satisfying λ ≤ |μ_{•}| is necessarily a weight for V_{|μ_{•}|}. Also, μ minuscule implies that Ω(V_{μ}) = W μ. Thus, Lemma 6.3 implies Lemma 6.2.

In the same way, we get the following result, proving Proposition 1.4.

Proposition 6.4. Suppose λ ≤ |μ_{•}| is dominant. Then there exist dominant coweights μ′_{i} ≤ μ_{i} and elements w_{i} ∈ W (1 ≤ i ≤ r) such that λ = w_{1}μ′_{1} + ⋯ + w_{r}μ′_{r}. Consequently, the P-R-V conjecture implies that Rep(μ′_{•}, λ) holds. In particular, we have

Hecke(μ_{•}, λ) ⟹ Rep(μ′_{•}, λ).

As for the assertion Hecke(μ_{•}, λ) of Proposition 1.5, at this point one could appeal to Theorem 1.1, but we prefer to give a direct proof. Indeed, the assertion is obvious from the remark that m_{μ_{•}}(Q_{μ_{•}}) = Q̄_{|μ_{•}|}. This in turn follows from the surjectivity of m_{μ_{•}}, since μ_{i} minuscule for every i means Q̄_{μ_{i}} = Q_{μ_{i}} and thus Q̃_{μ_{•}} = Q_{μ_{•}}.

7. Spaltenstein-Springer varieties and Proposition 1.7

We begin with the statement of a crucial theorem of Spaltenstein [Sp]. Let V denote a k-vector space of dimension d, and let (d_{1}, …, d_{r}) denote an ordered r-tuple of nonnegative integers such that d_{1} + ⋯ + d_{r} = d. The r-tuple d_{•} determines a standard parabolic subgroup P ⊂ GL(V). Let us consider the variety of partial flags of type P:

P = {V_{•} = (V = V_{0} ⊃ V_{1} ⊃ ⋯ ⊃ V_{r} = 0) | dim(V_{i−1}/V_{i}) = d_{i}, 1 ≤ i ≤ r},

which is isomorphic to GL(V)/P.

For any nilpotent endomorphism T ∈ End(V), let P^{T} denote the closed subvariety of P consisting of partial flags V_{•} such that T stabilizes each V_{i}. This is simply the fiber over T of the partial Springer resolution

π : Ñ^{P} → N ,

where N ⊂ End(V) is the nilpotent cone, Ñ^{P} = {(T, V_{•}) ∈ N × P | V_{•} ∈ P^{T} }, and the morphism π is the obvious forgetful one.

Inside P^{T} we may consider the closed subvariety P_{min}^{T} consisting of partial flags V_{•} ∈ P^{T} such that

T acts by zero on V_{i−1}/V_{i} (1 ≤ i ≤ r).

We call such varieties P_{min}^{T} the Spaltenstein-Springer varieties.

We have the following fundamental result of N. Spaltenstein ([Sp], final Corollary).

Theorem 7.1 (Spaltenstein). The irreducible components of a Spaltenstein-Springer variety P_{min}^{T} all have the same dimension.

Now we turn to the proof of Proposition 1.7, which we will reduce to Theorem 7.1.

Recall that μ_{•} = (μ_{1}, …, μ_{r}), where each μ_{i} is a dominant minuscule coweight for GL_{n}. Working in the affine Grassmannian for GL_{n} and fixing y ∈ Q_{λ} ⊂ Q̄_{|μ_{•}|}, we want to show that all the irreducible components of m_{μ_{•}}^{−1}(y) have the same dimension. Without loss of generality, we may assume μ_{i} = (1^{d_{i}}, 0^{n−d_{i}}), where 1 ≤ d_{i} ≤ n − 1. Write d := d_{1} + ⋯ + d_{r}. Suppose the point y ∈ Q_{λ} corresponds to the O-lattice L ⊂ F^{n}. We have L ⊂ O^{n}. Let T denote the nilpotent endomorphism of the ddimensional k-vector space V := O^{n}/L induced by multiplication by t. Then the fiber m_{μ_{•}}^{−1}(y) consists of chains of k-vector spaces

O^{n} = L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{r} = L

satisfying • dim(L_{i−1}/L_{i}) = d_{i}, for each i,

• T preserves each L_{i},
• T acts by zero on each L_{i−1}/L_{i}.

In other words, m_{μ_{•}}^{−1}(y) is a Spaltenstein-Springer variety for the d-dimensional k-vector space V and the nilpotent operator T. Thus, Proposition 1.7 follows from Theorem 7.1.

8. Proofs of Proposition 1.8 and Theorem 1.2

Proof of Proposition 1.8: We need the following set-up in order to bring Proposition 1.7 into play. Write each coweight μ_{i} as a sum of dominant minuscule coweights

μ_{i} = ν_{i1} + ν_{i2} + ⋯

We consider the following diagram

(Q_{ν_{11}} ˜× ⋯) ˜× ⋯ ˜×(Q_{ν_{r1}} ˜× ⋯)
η

Q̄_{μ_{1}} ˜× ⋯ ˜×Q̄_{μ_{r}}

m_{μ•}
Q̄_{|μ_{•}|},

where η = m_{ν_{1•}} ˜× ⋯ ˜×m_{ν_{r•}}. The composition m_{ν_{••}} = m_{μ_{•}} ∘ η is just the usual forgetful morphism, once the domain is identified with the twisted product of the varieties Q_{ν_{ij}}. In particular, all three morphisms m_{μ_{•}}, η, and m_{ν_{••}} are semi-small, proper, birational, surjective, and locally trivial in the stratified sense. Moreover, the conclusion of Proposition 1.7 applies to m_{ν_{••}}. Note that |ν_{••}| = |μ_{•}|.

Let y ∈ Q_{λ} ⊂ Q̄_{μ_{•}}. Suppose C′ is an irreducible component of m_{μ_{•}}^{−1}(y) such that C′ ∩ Q_{μ′_{•}} is an open dense subset in C′, so that dim(C′ ∩ Q_{μ′_{•}}) = dim(C′). Let C be any irreducible component of η^{−1}(C′) which dominates C′. Then C ∩ η^{−1}(Q_{μ′_{•}}) is non-empty, hence is an open dense subset of C. Thus, using the semi-smallness and local triviality of η, we get

dim(C) = dim(C ∩ η^{−1}(Q_{μ′_{•}}))
≤ dim(η^{−1}(C′ ∩ Q_{μ′_{•}}))
≤ ⟨ρ, |ν_{••}| − |μ′_{•}|⟩ + dim(C′ ∩ Q_{μ′_{•}})
= ⟨ρ, |μ_{•}| − |μ′_{•}|⟩ + dim(C′).

Now C is an irreducible component of m_{ν_{••}}^{−1}(y), hence by Proposition 1.7 has dimension ⟨ρ, |μ_{•}| − λ⟩. Therefore

dim(C′) ≥ ⟨ρ, |μ′_{•}| − λ⟩,

and the desired equality follows from the semi-smallness of m_{μ_{•}}.

Proof of Theorem 1.2:

This follows immediately from Proposition 1.3, Proposition 1.8 and the remark that since Q_{μ_{•}} is open in Q̃_{μ_{•}}, any irreducible component C of the fiber m_{μ_{•}}^{−1}(y) which meets Q_{μ_{•}} has C ∩ Q_{μ_{•}} as a dense open subset. This completes the proof of Theorem 1.2.

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Mathematics Department University of Maryland College Park, MD 20742-4015 USA tjh@math.umd.edu