EQUIDIMENSIONALITY OF CONVOLUTION MORPHISMS AND APPLICATIONS TO SATURATION PROBLEMS

THOMAS J. HAINES

Abstract. Fix a split connected reductive group G over a field k, and a positive integer r. For any r-tuple of dominant coweights μ_{i} of G, we consider the restriction m_{μ_{•}} of the r-fold convolution morphism of Mirkovic-Vilonen [MV1, MV2] to the twisted product of affine Schubert varieties corresponding to μ_{•}. We show that if all the coweights μ_{i} are minuscule, then the fibers of m_{μ_{•}} are equidimensional varieties, with dimension the largest allowed by the semi-smallness of m_{μ_{•}}. We derive various consequences: the equivalence of the nonvanishing of Hecke and representation ring structure constants, and a saturation property for these structure constants, when the coweights μ_{i} are sums of minuscule coweights. This complements the saturation results of Knutson-Tao [KT] and Kapovich-Leeb-Millson [KLM]. We give a new proof of the P-R-V conjecture in the “sums of minuscules” setting. Finally, we generalize and reprove a result of Spaltenstein pertaining to equidimensionality of certain partial Springer resolutions of the nilpotent cone for GL_{n}.

1. Introduction

Let G be a split connected reductive group over a finite field 𝔽_{q}, with Langlands dual Ĝ = Ĝ(\overline{ℚ}_{ℓ}), where char(𝔽_{q}) = p and ℓ ≠ p is prime. The geometric Satake isomorphism of Mirkovic-Vilonen [MV2] establishes a geometric construction of Ĝ. More precisely, it identifies Ĝ with the automorphism group of the fiber functor of a certain Tannakian category. Letting F = 𝔽_{q}((t)) and O = 𝔽_{q}[[t]], the latter is the category P_{G(O)} of G(O)-equivariant perverse \overline{ℚ}_{ℓ}-sheaves F on the affine Grassmannian

Q = G(F)/G(O),

viewed as an ind-scheme over 𝔽_{q}. The fiber functor

F ↦ H^{∗}(Q, F)

takes P_{G(O)} to the category of graded finite-dimensional \overline{ℚ}_{ℓ}-vector spaces. In order to give P_{G(O)} a Tannakian structure, one needs to endow it with a tensor product with commutativity and associativity constraints. There are a few different ways to construct the tensor product (see especially [Gi], [MV1], and [Ga]). The present article will use the construction in [MV1], which is defined in terms of the convolution morphism

m_{μ_{•}} : \overline{Q}_{μ_{1}} ˜× ⋯ ˜×\overline{Q}_{μ_{r}} → \overline{Q}_{|μ_{•}|}.

Here the μ_{i} are dominant cocharacters of G indexing various G(O)-orbits Q_{μ_{i}} ⊂ Q (via the Cartan decomposition), |μ_{•}| := ∑_{i} μ_{i}, and the morphism m_{μ_{•}} forgets all but the last element in the twisted product (see section 2). The morphism m_{μ_{•}} is used to construct the r-fold convolution product in P_{G(O)}, as follows. Given G(O)-equivariant perverse sheaves F_{1}, …, F_{r}, supported on various closures \overline{Q}_{μ_{1}}, …, \overline{Q}_{μ_{r}}, there is a well-defined perverse “twisted external product” sheaf F_{1} ˜⊠ ⋯ ˜⊠F_{r} on the twisted product \overline{Q}_{μ_{1}} ˜× ⋯ ˜×\overline{Q}_{μ_{r}}; see section 2. Then the r-fold convolution product is defined by the proper push-forward on derived categories

Research supported by NSF grant DMS 0303605, and by a Sloan Research Fellowship.

F_{1} ∗ ⋯ ∗ F_{r} = m_{!}(F_{1} ˜⊠ ⋯ ˜⊠F_{r}).

For brevity, let us write K = G(O), a maximal compact subgroup of the loop group G(F). Zariski-locally the twisted product \overline{Q}_{μ_{1}} ˜× ⋯ ˜×\overline{Q}_{μ_{r}} is just the usual product and the morphism m_{μ_{•}} is given by

m_{μ_{•}} : (g_{1}K, g_{2}K, …, g_{r}K) ↦ g_{1}g_{2} ⋯ g_{r} K.

Using this one may check that under the sheaf-function dictionary à la Grothendieck, the tensor structure on P_{G(O)} corresponds to the usual convolution in the spherical Hecke algebra H_{q} = C_{c}(K\G(F)/K). This is the convolution algebra of compactly-supported \overline{ℚ}_{ℓ}-valued functions on G(F) which are bi-invariant under K, where the convolution product (also denoted ∗) is defined using the Haar measure which gives K volume 1. This is the reason why we call m_{μ_{•}} a convolution morphism.

The morphism m_{μ_{•}} is projective, birational, and semi-small and locally-trivial in the stratified sense; see [MV1], [NP] and §2.2 for proofs of these properties, and [H] for some further discussion. These properties are essential for the construction of the tensor product on P_{G(O)}.

As is well-known, the fibers of the morphism m_{μ_{•}} carry representation-theoretic information (see section 2.3). The purpose of this article is to establish a new equidimensionality property of these fibers in a very special situation, and then to extract some consequences of combinatorial and representation-theoretic nature. The main result is the following theorem. Let ρ denote the half-sum of the positive roots for G, and recall that the semi-smallness of m_{μ_{•}} means that for every y ∈ Q_{λ} ⊂ \overline{Q}_{|μ_{•}|}, the fiber over y satisfies the following bound on its dimension

dim(m_{μ_{•}}^{−1}(y)) ≤ \frac{1}{2}[dim(\overline{Q}_{|μ_{•}|}) − dim(\overline{Q}_{λ})] = ⟨ρ, |μ_{•}| − λ⟩.

Theorem 1.1 (Equidimensionality for minuscule convolutions). Let y ∈ Q_{λ} ⊂ \overline{Q}_{|μ_{•}|}. Suppose each coweight μ_{i} is minuscule. Then every irreducible component of the fiber m_{μ_{•}}^{−1}(y) has dimension ⟨ρ, |μ_{•}| − λ⟩.

Recall that a coweight μ is minuscule if ⟨α, μ⟩ ∈ {−1, 0, 1} for every root α. The following result is a corollary of the proof.

Corollary 1.2. If every μ_{i} is minuscule, then each fiber m_{μ_{•}}^{−1}(y) admits a paving by affine spaces.

The conclusions in Theorem 1.1 fail without the hypothesis that each μ_{i} is minuscule. Without that hypothesis, the dimension of the fiber can be strictly less than ⟨ρ, |μ_{•}|−λ⟩. This can happen even if we weaken the hypothesis to “each μ_{i} is minuscule or quasi-minuscule”, see Remark 4.3. Further, even for G = GL_{n} there exist coweights of the form μ_{i} = (d_{i}, 0^{n−1}) where d_{1} + ⋯ + d_{r} = n, for which certain fibers m_{μ_{•}}^{−1}(y) are not equidimensional, see Remark 8.3. We do not know how to characterize the tuples μ_{•} for which every fiber m_{μ_{•}}^{−1}(y) is paved by affine spaces, see Question 3.9.

Nevertheless, a similar equidimensionality statement continues to hold when we require each μ_{i} to be a sum of minuscules (see §4). In its most useful form it concerns the intersection of the fiber m_{μ_{•}}^{−1}(y) with the open stratum Q_{μ_{•}} = Q_{μ_{1}} ˜× ⋯ ˜×Q_{μ_{r}} of the twisted product Q̃_{μ_{•}}. The following result is an easy corollary of Theorem 1.1. It is proved in Proposition 4.1 (see also [H], §8).

Theorem 1.3 (Equidimensionality for sums of minuscules). Suppose each μ_{i} is a sum of dominant minuscule coweights. Then the intersection

m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}}

is equidimensional of dimension ⟨ρ, |μ_{•}| − λ⟩, provided the intersection is non-empty.

This result also generally fails to hold without the hypothesis on the coweights μ_{i} (see Remark 4.3). Note that Theorem 1.1 is actually a special case of Theorem 1.3.

Theorem 1.3 allows us to establish a relation between structure constants of Hecke and representation rings, generalizing [H], which treated the case of GL_{n}. Namely, thinking of (μ_{•}, λ) as an r + 1-tuple of dominant weights of Ĝ (resp. coweights of G), we may define structure constants dim(V_{μ_{•}}^{λ}) (resp. c_{μ_{•}}^{λ}(q)) for the representation ring of the category Rep(Ĝ) (resp. for the Hecke algebra H_{q}) corresponding to the multiplication of basis elements consisting of highest-weight representations V_{μ_{1}}, …, V_{μ_{r}} (resp. characteristic functions f_{μ_{1}} = 1_{Kμ_{1}K}, …, f_{μ_{r}} = 1_{Kμ_{r}K}). In other words, we consider the decompositions

V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}} = ⨁_{λ} V_{μ_{•}}^{λ} ⊗ V_{λ}
f_{μ_{1}} ∗ ⋯ ∗ f_{μ_{r}} = ∑_{λ} c_{μ_{•}}^{λ}(q) f_{λ}

in Rep(Ĝ) and H_{q}, respectively. Following [H], consider the properties

Rep(μ_{•}, λ) : dim(V_{μ_{•}}^{λ}) > 0
Hecke(μ_{•}, λ) : c_{μ_{•}}^{λ}(q) ≠ 0.

It is a general fact that Rep(μ_{•}, λ) ⇒ Hecke(μ_{•}, λ), for all groups G (see [KLM], Theorem 1.13, and Corollary 2.4 below). The reverse implication holds for GL_{n}, but fails for general tuples μ_{•} attached to other groups (see [KLM],[H], and Remark 4.3). The following consequence of Theorem 1.3 shows that there is a natural condition on the coweights μ_{i} which ensures that the reverse implication does hold.

Theorem 1.4 (Equivalence of non-vanishing of structure constants). If each μ_{i} is a sum of dominant minuscule coweights of G, then

RepĜ(μ_{•}, λ) ⇔ Hecke^{G}(μ_{•}, λ).

Since every coweight of GL_{n} is a sum of minuscule coweights, this puts the GL_{n} case into a broader context. For groups not of type A, many (or all) coweights are not sums of minuscules, and this is reflected by the abundance of counterexamples to the implication Hecke(μ_{•}, λ) ⇒ Rep(μ_{•}, λ) for those groups.

As first pointed out by M. Kapovich, B. Leeb, and J. Millson [KLM], the translation from the representation ring structure constants to Hecke algebra structure constants has some applications, in particular to saturation questions for general groups. The authors of [KLM] investigated saturation questions for the structure constants of H_{q}, and their results apply to general groups G. Results such as Theorem 1.4 allow us to deduce saturation theorems for Rep(Ĝ).

Theorem 1.5 (A saturation theorem for sums of minuscules). Suppose μ_{•} is an r-tuple of dominant weights for Ĝ, whose sum belongs to the root lattice of Ĝ. Suppose each μ_{i} is a sum of dominant minuscule weights. Let V_{μ_{i}} denote the irreducible Ĝ-module with highest weight μ_{i}. Then

(1) If k = k_{G} denotes the Hecke algebra saturation factor for G as defined in [KLM], then

(V_{Nμ_{1}} ⊗ ⋯ ⊗ V_{Nμ_{r}})Ĝ ≠ 0 ⇒ (V_{kμ_{1}} ⊗ ⋯ ⊗ V_{kμ_{r}})Ĝ ≠ 0,

for every positive integer N.

(2) If the simple factors of G_{ad} are all of type A, B, C or E_{7}, then the above implication holds with k replaced by 1.

The analogue of part (1) for Hecke algebra structure constants is due to M. Kapovich, B. Leeb, and J. Millson [KLM]. We derive part (1) from their result by applying Theorem 1.4, with λ = 0. In fact a sharper version of part (1) is valid: we need only assume that at least r − 1 of the weights μ_{i} are sums of minuscules, see Theorem 7.2.

A somewhat more comprehensive version of part (2) is proved in Theorem 7.4, again by establishing the Hecke algebra analogue. That analogue is proved in Theorem 9.7 of the Appendix, written jointly with M. Kapovich and J. Millson. Based on this result and some computer calculations done using LiE, we conjecture that the conclusion of part (2) holds in all cases (i.e. factors of type D and E_{6} should also be allowed; see Conjecture 7.3).

Note that for Ĝ = GL_{n}(ℂ), part (2) is not new. It is the well-known saturation property of GL_{n}, which was first proved by A. Knutson and T. Tao in their paper [KT]. The Hecke algebra approach was introduced in [KLM], which provided a new proof of the Knutson-Tao result, and suggested that saturation problems for more general groups are best approached via Hecke algebras and triangles in Bruhat-Tits buildings.

In their recent preprint [KM], Kapovich and Millson have announced some results which are closely related to our Theorems 1.4 and 1.5, and which are proved by completely different methods; see Remarks 5.2, 7.5.

Theorems 1.1, 1.3, and 1.4 were proved for GL_{n} in [H], as consequences of the geometric Satake isomorphism, the P-R-V property, and Spaltenstein’s theorem in [Sp] on the equidimensionality of certain partial Springer resolutions. In this paper, the geometric Satake isomorphism (more precisely, a corollary of it, Theorem 2.2) remains a key ingredient, and in some sense this work could be viewed as an application of that powerful result. On the other hand, the present proofs of Theorems 1.1-1.4 rely on neither the P-R-V property nor Spaltenstein’s theorem. In fact, here we turn the logic around, giving a new proof of the P-R-V property in the “sums of minuscules” situation, and also giving a new proof and a generalization of Spaltenstein’s theorem. Those results are explained in sections 6 and 8, respectively.

Acknowledgments. I express my thanks to Misha Kapovich and John Millson for generously sharing their ideas with me, especially in relation to the Appendix, which was written jointly with them. I also thank them for giving me early access to their recent work [KM]. I am indebted to Jeff Adams for his invaluable help with LiE; his programs were used to run some extensive computer checks of Conjecture 7.3. Finally, I thank the referee for some very helpful suggestions for simplifying the proof of Theorem 3.1.

2. Preliminaries and notation

2.1. General notation. Let k denote a field, usually taken to be the complex numbers ℂ, a finite field 𝔽_{q}, or an algebraic closure \overline{𝔽}_{q} of a finite field. Let O = k[[t]] (resp. F = k((t))) denote the ring of formal power series (resp. Laurent series) over k.

Let G denote a split connected reductive group over k. Fix a k-split maximal torus T and a k-rational Borel subgroup B containing T. We have B = T U, where U is the unipotent radical of B. Let X_{+} ⊂ X_{∗}(T) denote the set of B-dominant integral coweights for G. By W we denote the finite Weyl group N_{G}(T)/T. The Bruhat order ≤ on W will always be the one determined by the Borel B we have fixed. Let w_{0} denote the longest element in W.

Consider the “loop group” G(F) = G(k((t))) as an ind-scheme over k. Occasionally we designate this by LG, and the “maximal compact” subgroup G(O) by L^{≥0}G or simply K. The affine Grassmannian Q (over the field k) is the fpqc-quotient sheaf G(k((t)))/G(k[[t]]); it is an ind-scheme. If G = GL_{n} and R is a k-algebra, Q(R) is the set of all R[[t]]-lattices in R((t))^{n}. If G = GSp_{2n}, it is the set of lattices in R((t))^{2n} which are self-dual up to an element in R[[t]]^{×}.

By the Cartan decomposition we have a stratification into G(k[[t]])-orbits:

Q = ∐_{μ∈X_{+}} G(k[[t]])μG(k[[t]])/G(k[[t]]).

Here we embed X_{∗}(T) into G(k((t))) by the rule μ ↦ μ(t) ∈ T (k((t))). We will denote the G(k[[t]])-orbit of μ simply by Q_{μ} in the sequel. The closure relations are determined by the standard partial order ⪯ on dominant coweights: Q_{λ} ⊂ \overline{Q}_{μ} if and only λ ⪯ μ, which by definition holds if and only if μ − λ is a sum of B-positive coroots. Given L, L′ ∈ Q, let inv(L, L′) ∈ X_{+} denote the relative position of L, L′, where by definition

inv(gK, g′K) = λ ⇔ g^{−1}g′ ∈ KλK.

There is a canonical perfect pairing ⟨· , ·⟩ : X^{∗}(T) × X_{∗}(T) → ℤ. Let ρ denote the halfsum of the B-positive roots of G. Given μ ∈ X_{+}, the K-orbit Q_{μ} is a smooth quasiprojective variety of dimension ⟨2ρ, μ⟩ over k. Let \overline{Q}_{μ} ⊂ Q denote the closure of Q_{μ} in the ind-scheme Q.

Let e_{0} denote the base point in the affine Grassmannian for G, i.e., the point corresponding to the coset K ∈ G(F)/K. For ν ∈ X_{∗}(T), let t_{ν} := ν(t) ∈ LG. For a dominant coweight λ, denote e_{λ} = t_{λ}e_{0}.

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

Q̃_{μ_{•}} = \overline{Q}_{μ_{1}} ˜× ⋯ ˜×\overline{Q}_{μ_{r}}

to be the subscheme of Q^{r} 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}). The projection onto the last coordinate gives the proper surjective birational morphism

m_{μ_{•}} : Q̃_{μ_{•}} → \overline{Q}_{|μ_{•}|},

where by definition |μ_{•}| = ∑_{i} μ_{i}.

Note that the target of m_{μ_{•}} is stratified by the K-orbits Q_{λ} for λ ranging over dominant coweights satisfying λ ⪯ |μ_{•}|. Similarly, the domain is stratified by the locally closed twisted products Q_{μ′_{•}} := Q_{μ′_{1}} ˜× ⋯ ˜×Q_{μ′_{r}}, where μ′_{i} ranges over dominant coweights satisfying μ′_{i} ⪯ μ_{i}. Here Q_{μ′_{•}} is defined exactly as is Q̃_{μ′_{•}}, except that the conditions inv(L_{i−1}, L_{i}) ⪯ μ′_{i} are replaced with inv(L_{i−1}, L_{i}) = μ′_{i}.

With respect to these stratifications, m_{μ_{•}} is locally trivial and semi-small (in the stratified sense). The local triviality is discussed in §2.2. The semi-smallness means that for every inclusion Q_{λ} ⊂ m_{μ_{•}}(Q_{μ′_{•}}), the fibers of the restricted morphism

m_{μ_{•}} : m_{μ_{•}}^{−1}(Q_{λ}) ∩ Q_{μ′_{•}} → Q_{λ}

have dimension bounded above by

\frac{1}{2}[dim(Q_{μ′_{•}}) − dim(Q_{λ})] = ⟨ρ, |μ′_{•}| − λ⟩.

When we work in the context of Hecke algebras H_{q}, the field k will be the finite field 𝔽_{q}, where q = p^{j} for a prime p. In any case, we will always fix a prime ℓ ≠ char(k), and fix an algebraic closure \overline{ℚ}_{ℓ} of ℚ_{ℓ}. We define the dual group Ĝ = Ĝ(\overline{ℚ}_{ℓ}). We let T̂ ⊂ Ĝ denote the dual torus of T, defined by the equality X^{∗}(T̂) = X_{∗}(T).

Let Q^{∨} = Q^{∨}(G) (resp. Q = Q(G)) denote the lattice in X_{∗}(T) (resp. X^{∗}(T)) spanned by the coroots (resp. roots) of G in T. There is a canonical identification Q^{∨}(G) = Q(Ĝ), by which we can define a notion of simple positive root in Ĝ and thus a corresponding Borel subgroup B̂ containing T̂.

When we consider an r + 1-tuple of coweights (μ_{•}, λ), it will always be assumed that ∑_{i} μ_{i} − λ ∈ Q^{∨}. (When thinking of these as weights of T̂, this amounts to assuming that ∑_{i} μ_{i} − λ ∈ Q(Ĝ).)

For μ dominant we let Ω(μ) denote the set of weights of the irreducible representation of Ĝ with highest weight μ. For ν ∈ X_{∗}(T), we let S_{ν} = Ut_{ν}e_{0}.

If μ is dominant then we denote by V_{μ} the irreducible Ĝ-module with highest weight μ. Its contragredient (V_{μ})^{∗} is also irreducible, so we can define the dual dominant coweight μ^{∗} by the equality V_{μ^{∗}} = (V_{μ})^{∗}. We have μ^{∗} = −w_{0}μ.

We shall make frequent use of the fact that S_{ν} ∩ \overline{Q}_{μ} ≠ ∅ only if ν ∈ Ω(μ) ([BT], 4.4.4, or [NP], Lemme 4.2). For any ν ∈ X_{∗}(T), let ν_{d} denote the unique B-dominant element in W ν. The Weyl group permutes the set of (co)weights, and we let W_{μ} denote the stabilizer in W of μ.

Recall that a coweight μ is minuscule provided that ⟨α, μ⟩ ∈ {−1, 0, 1}, for every root α. Viewing μ as a weight of Ĝ, this is equivalent to the statement that Ω(V_{μ}) = W μ (see [Bou]).

2.2. Local triviality of the morphism m_{μ_{•}}. Let X = ∪_{i}X_{i} and Y = ∪_{j}Y_{j} be stratifications of algebraic varieties over k by locally closed subvarieties, having the property that the boundary of any stratum is a union of other strata.

Suppose we have a morphism f : X → Y. We suppose that f is proper and that each f(X_{i}) is a union of strata Y_{j}. We say f is locally trivial in the stratified sense, if for every y ∈ Y_{j} there is a Zariski-open subset V ⊂ Y_{j} with y ∈ V, and a stratified variety F, such that there is an isomorphism of stratified varieties

(2.2.1)  f^{−1}(V) ≅ F × V

which commutes with the projections to V. ^{1}

The following lemma is well-known, see [MV1]. We give the proof for the convenience of the reader.

Lemma 2.1. The morphism m_{μ_{•}} is Zariski-locally trivial in the stratified sense.

Proof. Fix y ∈ Q_{λ} ⊂ \overline{Q}_{|μ_{•}|}. We can identify Q_{λ} with the quotient in the notation of loop groups

Q_{λ} = L^{≥0}G/L^{≥0}G ∩ L^{≥λ}G,

1This definition differs from that used in [H]. In that paper, a weaker notion of “locally trivial in the stratified sense” was used. This paper requires the present (more conventional) definition.

where by definition L^{≥λ}G = λL^{≥0}Gλ^{−1}. Suppose that Zariski-locally on the base, the projection

(2.2.2)  L^{≥0}G → L^{≥0}G/L^{≥0}G ∩ L^{≥λ}G

has a section. Then it is easy to see that Zariski-locally, there is an isomorphism as in (2.2.1) for f = m_{μ_{•}}. Indeed, suppose L_{•} = (L_{1}, …, L_{r}) ∈ Q̃_{μ_{•}} has L_{r} ∈ Q_{λ}. Then for L_{r} in a Zariski-neighborhood V of e_{λ} in Q_{λ}, we can write L_{r} = ke_{λ} for a well-defined k ∈ L^{≥0}G, the image of L_{r} under the local section. Then we may define (2.2.1) by

(L_{1}, …, L_{r}) ↦ (k^{−1}L_{1}, …, k^{−1}L_{r}) × L_{r}.

It remains to prove that (2.2.2) is Zariski-locally trivial. By [NP], Lemme 2.3, we can write L^{≥0}G∩L^{≥λ}G = P_{λ} ⋉(L^{>0}G∩L^{≥λ}G), where P_{λ} ⊂ G is the parabolic subgroup corresponding to the roots α such that ⟨α, λ⟩ ≤ 0, and where L^{>0}G is the kernel of the morphism L^{≥0}G → G induced by t ↦ 0. We also have an obvious isomorphism L^{≥0}G = G ⋉ L^{>0}G. Then (2.2.2) can be factored as the composition of two projections

(2.2.3)  L^{≥0}G → L^{≥0}G/L^{>0}G ∩ L^{≥λ}G = G × [L^{>0}G/L^{>0}G ∩ L^{≥λ}G],

and

(2.2.4)  G × [L^{>0}G/L^{>0}G ∩ L^{≥λ}G] → L^{≥0}G/L^{≥0}G ∩ L^{≥λ}G.

Here the first projection is the obvious one, and the second projection is the quotient for the right action of P_{λ} on G × [L^{>0}G/L^{>0} ∩ L^{≥λ}G] given by

(g, g^{+}L^{>0}G ∩ L^{≥λ}G) · p = (gp, p^{−1}g^{+}pL^{>0}G ∩ L^{≥λ}G).

The morphism (2.2.3) is actually trivial, because the multiplication map

(L^{>0}G ∩ L^{<λ}G) × (L^{>0}G ∩ L^{≥λ}G) → L^{>0}G

is an isomorphism, where L^{<λ}G := λL^{<0}Gλ^{−1} and where L^{<0}G denotes the kernel of the map G(k[t^{−1}]) → G induced by t^{−1} ↦ 0; see [NP], §2.

The morphism (2.2.4) has local sections in the Zariski topology, coming from the embedding of the “big cell” U\overline{P_{λ}} ↪ G/P_{λ}. This completes the proof. □

2.3. Review of information carried by fibers of convolution morphisms. The following well-known result plays a key role in this article.

Theorem 2.2 (Geometric Satake Isomorphism – weak form). For every tuple (μ_{•}, λ), and every y ∈ Q_{λ}, there is an equality

dim(V_{μ_{•}}^{λ}) = # irreducible components of m_{μ_{•}}^{−1}(y) having dimension ⟨ρ, |μ_{•}| − λ⟩.

See [H], §3 for the proof of this assuming the geometric Satake isomorphism in the context of finite residue fields. We also have the following elementary lemma.

Lemma 2.3. For (μ_{•}, λ), y as above,

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

For context we recall following [H] that the above two statements together with the Weil conjectures yield the following expression for the Hecke algebra structure constants.

Corollary 2.4 ([KLM]). With μ_{•}, λ as above, the Hecke algebra structure constant is given by the formula

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

This formula was first proved by Kapovich, Leeb, and Millson [KLM], who deduced it from the results of Lusztig [Lu2]. It actually provides an algorithm to compute the multiplicities dim(V_{μ_{•}}^{λ}). Indeed, one can determine the polynomial c_{μ_{•}}^{λ}(q) by computing products in an Iwahori-Hecke algebra, using the Iwahori-Matsumoto presentation of that algebra. Of course this involves the computation of much more than just the leading term of c_{μ_{•}}^{λ}, so in practice this procedure is not a very efficient way to compute dim(V_{μ_{•}}^{λ}).

However the formula does make it clear that the dimensionality of the fiber m_{μ_{•}}^{−1}(y) plays a role in linking the non-vanishing of the structure constants: if dim(V_{μ_{•}}^{λ}) > 0, then evidently c_{μ_{•}}^{λ}(q) ≠ 0 for all large q (and thus all q, by the argument in [H], §4). On the other hand, if c_{μ_{•}}^{λ}(q) ≠ 0, it could well happen that the leading coefficient dim(V_{μ_{•}}^{λ}) is zero. However, if we knew a priori that whenever the space m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}} is non-empty, it is actually of dimension ⟨ρ, |μ_{•}| − λ⟩, then the non-vanishing of c_{μ_{•}}^{λ}(q) would imply the non-vanishing of its leading coefficient. We will prove this dimension statement for m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}} in the case where each μ_{i} is a sum of minuscules, by a reduction to the case where each μ_{i} is minuscule. But as is seen in the reduction step (the “pulling apart” Lemma 4.2) it is necessary to prove the stronger fact that in that case, the fibers are not just of largest possible dimension, but are also equidimensional.

Our first goal, therefore, is to establish the (equi)dimensionality statement just mentioned (in Theorem 3.1 below). Let us first pause to mention some related work in the literature. After the seminal work of Mirkovic-Vilonen [MV1, MV2] on which everything else is based, the author was particularly inspired by the work of Ngô-Polo [NP]. Many other authors have had the idea to use the fibers of the morphisms m_{μ_{•}} to derive representation-theoretic consequences, and the works of Gaussent-Littelmann [GL] and of J. Anderson [A] seem particularly related to the present one. In fact, in [A] Anderson independently observed the relation between fibers of convolution morphisms and MV cycles (loc. cit. Theorem 8), which was the starting point in the proof of our Theorem 3.1.

3. Equidimensionality of minuscule convolutions

3.1. Proof of the main theorem. For this section we fix an r-tuple μ_{•} = (μ_{1}, …, μ_{r}) such that each μ_{i} is dominant and minuscule. The main result of this paper is the following theorem.

Theorem 3.1. The fibers of the morphism

m_{μ_{•}} : Q̃_{μ_{•}} → \overline{Q}_{|μ_{•}|}

are equidimensional. More precisely, if y ∈ Q_{λ} ⊂ \overline{Q}_{|μ_{•}|}, then every irreducible component of m_{μ_{•}}^{−1}(y) has dimension ⟨ρ, |μ_{•}| − λ⟩.

We will prove the theorem by induction on r, the number of elements in the tuple μ_{•} (the case of r = 1 being trivial). Let us suppose the theorem is true for the morphism

m_{μ′_{•}} : Q_{μ_{1}} ˜× ⋯ ˜×Q_{μ_{r−1}} → \overline{Q}_{|μ′_{•}|}

attached to the (r − 1)-tuple μ′_{•} = (μ_{1}, …, μ_{r−1}). We fix an orbit Q_{λ} ⊂ \overline{Q}_{|μ_{•}|} and a point y ∈ Q_{λ}; we want to prove the equidimensionality of the variety m_{μ_{•}}^{−1}(y). By equivariance under the K-action, we can assume that y = e_{λ} := t_{λ}e_{0}. We suppose that (L_{1}, …, L_{r−1}, L_{r}) ∈ m_{μ_{•}}^{−1}(y).

Since the relative position inv(L_{r−1}, L_{r}) = μ_{r} and L_{r} = e_{λ}, we deduce that L_{r−1} ∈ t_{λ}Kt_{μ}e_{0}, where μ := μ_{r}^{∗}. Thus, L_{r−1} ranges over the set

\overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ}.

Most of the work in the proof of Theorem 3.1 involves the attempt to understand this set. It is hard to understand the whole set, but as we shall see below, we can exhaust it by locally closed subsets Z_{w} which are easier to understand. The locally closed subsets help us compute dimensions of components in m_{μ_{•}}^{−1}(y) because, as we shall see,

• the morphism m_{μ′_{•}} becomes trivial over each of these subsets, and

• we can explicitly calculate the dimensions of the subsets. More precisely, since μ is a minuscule coweight, we decompose t_{λ}Q_{μ} as the union of the

locally closed subsets

t_{λ}( Ut_{wμ}e_{0} ∩ Kt_{μ}e_{0}) = Ut_{λ+wμ}e_{0} ∩ t_{λ}Kt_{μ}e_{0},

where w ∈ W/W_{μ}.

Using the above decomposition of t_{λ}Q_{μ}, we see that \overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ} is the disjoint union of the following locally-closed subvarieties

Z_{w} := \overline{Q}_{|μ′_{•}|} ∩ t_{λ}(S_{wμ} ∩ Q_{μ}),

where we recall that S_{ν} := Ut_{ν}e_{0} for ν ∈ X_{∗}(T). For brevity, we let X_{w} := S_{wμ} ∩ Q_{μ}, so that Z_{w} = \overline{Q}_{|μ′_{•}|} ∩ t_{λ}X_{w}.

We begin the proof with a preliminary lemma concerning X_{w}, and then proceed to some lemmas concerning Z_{w}.

The following result is due to Ngô-Polo [NP]. In this statement R^{+} denotes the set of positive roots relative to the Borel subgroup B = T U.

Lemma 3.2. Let ν be and dominant coweight, and let w ∈ W. Then there is an isomorphism of varieties

S_{wν} ∩ Q_{ν} = ∏_{α∈R^{+}∩wR^{+}} ∏_{i=0}^{⟨w^{−1}α,ν⟩−1} U_{α,i}t_{wν}e_{0},

where U_{α} ⊂ LU is the root subgroup corresponding to the positive root α, and U_{α,i} consists of the elements in U_{α} of form u_{α}(xt^{i}), x ∈ k, where u_{α} : 𝔾_{a} → U_{α} is the root homomorphism for α.

In particular, applying this to ν = μ, we see that X_{w} is an affine space of dimension

dim(X_{w}) = ⟨ρ, μ + wμ⟩.

Proof. The isomorphism is proved in [NP], Lemme 5.2. The dimension formula then follows, using the formula

ρ + w^{−1}ρ = ∑_{α∈R^{+}∩w^{−1}R^{+}} α,

and its consequence

⟨ρ, μ + wμ⟩ = ∑_{α∈R^{+}∩wR^{+}}⟨α, wμ⟩.

□

Consider next the morphism

p : m_{μ_{•}}^{−1}(t_{λ}e_{0}) ↠ \overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ}

given by p(L_{1}, …, L_{r−1}, L_{r}) = L_{r−1}.

Lemma 3.3. Suppose Z_{w} ≠ ∅. Then t_{λ+wμ}e_{0} ∈ Z_{w}, and the morphism p : p^{−1}(Z_{w}) ↠ Z_{w} is trivial. In particular

(3.1.1)  p^{−1}(Z_{w}) ≅ m_{μ′_{•}}^{−1}(t_{λ+wμ}e_{0}) × Z_{w}.

Proof. Note that

\overline{Q}_{|μ′_{•}|} ∩ t_{λ}(Ut_{wμ}e_{0} ∩ Kt_{μ}e_{0}) ≠ ∅ ⇒ \overline{Q}_{|μ′_{•}|} ∩ Ut_{λ+wμ}e_{0} ≠ ∅
⇒ λ + wμ ∈ Ω(|μ′_{•}|)
⇒ t_{λ+wμ}e_{0} ∈ \overline{Q}_{|μ′_{•}|}.

Since clearly t_{λ+wμ}e_{0} ∈ t_{λ}(S_{wμ} ∩ Q_{μ}), the first statement follows.

Now we prove the second statement. By Lemma 3.2, an element in t_{λ}(Ut_{wμ}e_{0} ∩Kt_{μ}e_{0}) can be written uniquely in the form

(3.1.2)  t_{λ}( ∏_{⟨α,wμ⟩=1^{α∈R^{+}}} u_{α}(x_{α})t_{wμ}e_{0}),

where x_{α} ∈ k for all α. Because λ is dominant, we can write this as

(3.1.3)  n_{0}t_{λ+wμ}e_{0},

for some uniquely determined n_{0} ∈ L^{≥0}U. Therefore if (L_{1}, …, L_{r−1}, L_{r}) ∈ p^{−1}(Z_{w}), we can write L_{r−1} = n_{0}t_{λ+wμ}e_{0} for a uniquely determined n_{0} ∈ L^{≥0}U. Then the isomorphism p^{−1}(Z_{w}) → m_{μ′_{•}}^{−1}(t_{λ+wμ}e_{0}) × Z_{w} is the map sending

(L_{1}, …, L_{r−1}, L_{r}) ↦ (n_{0}^{−1}L_{1}, …, n_{0}^{−1}L_{r−1}) × L_{r−1}.

Note that since m_{μ′_{•}} is K-equivariant and n_{0} ∈ K, we also have an isomorphism

p^{−1}(Z_{w}) ≅ m_{μ′_{•}}^{−1}(L_{r−1}) × Z_{w},

for any L_{r−1} ∈ Z_{w}. □

Lemma 3.4. (1) We have t_{λ}X_{w} ⊂ Q_{(λ+wμ)_{d}}.

(2) We have Z_{w} ≠ ∅ ⟺ (λ + wμ)_{d} ≤ |μ′_{•}|, in which case Z_{w} = t_{λ}X_{w}.

Proof. (1): This follows immediately from equations (3.1.2) and (3.1.3) above. (2): Clearly since t_{λ}X_{w} ⊂ Q_{(λ+wμ)_{d}}, we have Z_{w} ≠ ∅ if and only if Q_{(λ+wμ)_{d}} ⊂ \overline{Q}_{|μ′_{•}|}, which holds if and only if (λ + wμ)_{d} ≤ |μ′_{•}|. It is also clear that Z_{w} = t_{λ}X_{w} in that case. □

Recall that p^{−1}(Z_{w}) ⊂ m_{μ_{•}}^{−1}(e_{λ}). By the semi-smallness of m_{μ_{•}}, for every w we have dim(p^{−1}(Z_{w})) ≤ ⟨ρ, |μ_{•}|−λ⟩. We call Z_{w} good if equality holds. If Z_{w} is good, then p^{−1}(Z_{w}) is equidimensional of dimension ⟨ρ, |μ_{•}|−λ⟩. If Z_{w} is not good, then p^{−1}(Z_{w}) is a equidimensional of strictly smaller dimension. (We already know using the induction hypothesis and triviality that p^{−1}(Z_{w}) is equidimensional for every w.)

We need to give a concrete criterion for “Z_{w} is good”.

Lemma 3.5. Suppose Z_{w} ≠ ∅. Then Z_{w} is good if and only if λ + wμ is dominant.

Proof. Using (3.1.1) we have

dim(Z_{w}) + dim(m_{μ′_{•}}^{−1}(t_{(λ+wμ)_{d}}e_{0})) = dim(p^{−1}(Z_{w})) ≤ ⟨ρ, |μ_{•}| − λ⟩,

with equality if and only if Z_{w} is good. Since Z_{w} is non-empty, we know by Lemma 3.4 and Lemma 3.2 that dim(Z_{w}) = ⟨ρ, μ + wμ⟩. Using this together with our induction hypothesis that dim(m_{μ′_{•}}^{−1}(t_{(λ+wμ)_{d}}e_{0})) = ⟨ρ, |μ′_{•}| − (λ + wμ)_{d}⟩, and the equality ⟨ρ, |μ_{•}|⟩ = ⟨ρ, |μ′_{•}| + μ⟩, the above statement becomes

⟨ρ, (λ + wμ) − (λ + wμ)_{d}⟩ ≤ 0

with equality if and only if Z_{w} is good.

Thus we see that λ + wμ = (λ + wμ)_{d} if and only if Z_{w} is good. □

Lemma 3.6. For any w ∈ W, let \overline{Z_{w}} denote the closure of Z_{w} in \overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ}. Suppose Z_{w} and Z_{w′} are non-empty. Then Z_{w} ⊂ \overline{Z_{w′}} if w′ ≤ w in the Bruhat order on W.

Proof. Let P^{−} denote the standard parabolic determined by the set of roots satisfying ⟨α, μ⟩ ≤ 0. By Lemma 3.4, we have Z_{w′} = t_{λ}X_{w′} and Z_{w} = t_{λ}X_{w}, so that the closure relations for the Z_{w}’s inside t_{λ}Q_{μ} are determined by those for the X_{w}’s inside Q_{μ}.

By Lemma 3.2 the “reduction modulo t” isomorphism Q_{μ} ˜→ G/P^{−} induces an isomorphism

X_{w} ˜→ UwP^{−}/P^{−}.

(comp. [NP], Lemme 6.2.) The result now follows from the relation between the Bruhat order on W and the closure relations for U-orbits in G/P^{−}. □

Proof of Theorem 3.1. Consider again the morphism

p : m_{μ_{•}}^{−1}(t_{λ}e_{0}) → \overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ}

given by p(L_{1}, …, L_{r−1}, L_{r}) = L_{r−1}. We have constructed a decomposition of the range by locally closed sets Z_{w}, w ∈ W, over which p is trivial. Some of the sets Z_{w} might be empty, but for non-empty Z_{w}, we now have a useful description of those which are good (Lemma 3.5). A priori we do not know whether any good subsets exist, but in the course of the proof we shall see that they do.

Using our induction hypothesis, we know that for good Z_{w}, the set \overline{p^{−1}(Z_{w})} is a union of irreducible components of m_{μ_{•}}^{−1}(t_{λ}e_{0}) having dimension ⟨ρ, |μ_{•}| − λ⟩. It remains to prove that for any non-empty set Z_{w} which is not good, there exists a non-empty good Z_{w^{∗}} such that p^{−1}(Z_{w}) ⊂ \overline{p^{−1}(Z_{w^{∗}})}.

The first step is to find a good Z_{w^{∗}} such that Z_{w} ⊂ \overline{Z_{w^{∗}}}. Assume Z_{w} is non-empty but is not good. Then let w^{∗} be the unique element of minimal length in the subset W_{λ}wW_{μ} of W. Since λ + w^{∗}μ and λ + wμ are W_{λ}-conjugate, we have (λ + w^{∗}μ)_{d} = (λ + wμ)_{d} and hence by Lemma 3.4, Z_{w^{∗}} ≠ ∅. Then by Lemma 3.6 we have Z_{w} ⊂ \overline{Z_{w^{∗}}}. It remains to prove Z_{w^{∗}} is good, i.e. that λ + w^{∗}μ is dominant (Lemma 3.5). But if λ + w^{∗}μ is not dominant, there is a positive root α with ⟨α, λ + w^{∗}μ⟩ < 0. Since λ is dominant and w^{∗}μ is minuscule, we must have ⟨α, λ⟩ = 0 and ⟨α, w^{∗}μ⟩ = −1. The latter implies that (w^{∗})^{−1}α < 0, which means that s_{α}w^{∗} < w^{∗} in the Bruhat order on W. But since s_{α} ∈ W_{λ}, this contradicts the definition of w^{∗}.

To complete the proof of Theorem 3.1, we need to show that p^{−1}(Z_{w}) ⊂ \overline{p^{−1}(Z_{w^{∗}})}. Roughly, this follows because λ + wμ and λ + w^{∗}μ are W_{λ}-conjugate, hence both Z_{w} and Z_{w^{∗}} belong to Q_{λ+w^{∗}μ}, over which m_{μ′_{•}} is locally trivial (Lemma 2.1). More precisely, suppose L_{•} := (L_{1}, …, L_{r−1}, L_{r}) ∈ p^{−1}(Z_{w}). Write y ∈ Z_{w} for L_{r−1}, the image of (L_{1}, …, L_{r−1}) under m_{μ′_{•}}.

Let F = m_{μ′_{•}}^{−1}(y). Choose an open neighborhood y ∈ U ⊂ Q_{λ+w^{∗}μ} ∩ t_{λ}Q_{μ} over which m_{μ′_{•}} is trivial. Since p is the just the restriction of m_{μ′_{•}} over \overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ}, it follows that p is also trivial over U, so that p^{−1}(U) ≅ F × U.

To show

L_{•} ∈ \overline{p^{−1}(Z_{w^{∗}})},

it is enough to show

L_{•} ∈ p^{−1}(U) ∩ \overline{p^{−1}(Z_{w^{∗}})}.

The intersection on the right hand side contains

\overline{p^{−1}(U) ∩ p^{−1}(Z_{w^{∗}})} ≅ \overline{F × (U ∩ Z_{w^{∗}})}
= F × \overline{U ∩ Z_{w^{∗}}}
= F × U
≅ p^{−1}(U),

where for V open and A arbitrary, \overline{V ∩ A} denotes the closure of V ∩ A in the subspace topology on V. In proving the second equality we have used the fact that U ∩ Z_{w^{∗}} is nonempty and open in U, and that U is irreducible. These statements follow from the fact that the irreducible set Z_{w^{∗}} is open and dense in Z_{[w^{∗}]} := Q_{λ+w^{∗}μ} ∩ t_{λ}Q_{μ} (as proved in Lemma 3.7 below).

Our assertion now follows since L_{•} obviously belongs to p^{−1}(U). This completes the proof of Theorem 3.1, modulo Lemma 3.7 below. □

3.2. Description of closure relations, and paving by affine spaces. Lemma 3.6 gives a partial description of the closure relations between the Z_{w} subsets. Our present aim is to give a complete description.

Note that every class [w] ∈ W_{λ}\W/W_{μ} gives rise to a well-defined K-orbit Q_{λ+wμ}. Each double coset is represented by a unique element w^{∗} of minimal length. In other words, w^{∗} is the unique element of minimal length in its double coset W_{λ}w^{∗}W_{μ}. As remarked in the proof of Theorem 3.1, the coweight λ + w^{∗}μ is dominant.

If λ + w^{∗}μ ≤ |μ′_{•}|, we denote

Z_{[w^{∗}]} := Q_{λ+w^{∗}μ} ∩ t_{λ}Q_{μ} = ⋃_{w∈W_{λ}w^{∗}W_{μ}} Q_{λ+w^{∗}μ} ∩ t_{λ}(S_{wμ} ∩ Q_{μ});

in case λ + w^{∗}μ ≰ |μ′_{•}|, set Z_{[w^{∗}]} = ∅. Concerning the second equality defining Z_{[w^{∗}]}, it is clear that the left hand side contains the right hand side. To prove the other inclusion, note that if a subset of the left hand side of form Q_{λ+w^{∗}μ} ∩ t_{λ}(S_{wμ} ∩ Q_{μ}) is non-empty, then it is a Z_{w}, and has (λ + wμ)_{d} = λ + w^{∗}μ, from which it follows that w ∈ W_{λ}w^{∗}W_{μ}.

Clearly we have a decomposition by locally closed (possibly empty) subsets

\overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ} = ∐_{[w^{∗}]∈W_{λ}\W/W_{μ}} Z_{[w^{∗}]}.

Lemma 3.7. The following statements hold.

(a) For w ∈ W, let \overline{Z_{w}} denote the closure of Z_{w} in \overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ}. If Z_{w} ≠ ∅, then

\overline{Z_{w}} = ⋃_{v≥w} Z_{v}.

Furthermore, Z_{w} ≠ ∅ ⇒ Z_{v} ≠ ∅ for all v ≥ w.

(b) We have Z_{w^{∗}} ≠ ∅ if and only if Z_{w} ≠ ∅ for any w ∈ [w^{∗}].

(c) If Z_{w^{∗}} ≠ ∅, then the map t_{λ}Q_{μ} → Q_{μ} → G/P^{−} induces an isomorphism

Z_{[w^{∗}]} ˜⟶ ⋃_{w∈W_{λ}w^{∗}W_{μ}} UwP^{−}/P^{−},

and furthermore Z_{w^{∗}} is dense and open in Z_{[w^{∗}]}.

(d) Let v^{∗} denote a minimal representative for a double coset W_{λ}v^{∗}W_{μ}. If Z_{w} ≠ ∅ then

\overline{Z_{w}} ∩ Z_{[v^{∗}]} = ⋃_{v∈[v^{∗}], v≥w} Z_{v}.

Corollary 3.8. The irreducible components of \overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ} are the closures \overline{Z_{w^{∗}}}, where w^{∗} ranges over the minimal elements in the set {v^{∗} | λ + v^{∗}μ ≤ |μ′_{•}|}.

Proof. (a): The morphism

\overline{Q}_{|μ′_{•}|} ∩ t_{λ}Q_{μ} → G/P^{−}

is a closed immersion, hence proper. So if Z_{w} ≠ ∅, then the image of \overline{Z_{w}} is the closure

\overline{UwP^{−}/P^{−}} = ⋃_{v≥w} UvP^{−}/P^{−}.

It follows that Z_{w} ≠ ∅ ⇒ Z_{v} ≠ ∅, for all v ≥ w, and that the closure above is the image of

⋃_{v≥w} Z_{v}.

(b): This follows from Lemma 3.4, using the equality (λ + wμ)_{d} = λ + w^{∗}μ.

(c): This is easy, the main point being that Uw^{∗}P^{−}/P^{−} is clearly open and dense in the union of all UwP^{−}/P^{−} for w ∈ W with w ≥ w^{∗}.

(d): This follows from (a)-(c). □

Proof of Corollary 1.2. We can now prove that m_{μ_{•}}^{−1}(y) is indeed paved by affine spaces. Let us recall what this means. By definition, a scheme X is paved by affine spaces if there is an increasing filtration ∅ = X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} = X by closed subschemes X_{i} such that each successive difference X_{i}\X_{i−1} is a (topological) disjoint union of affine spaces 𝔸^{n_{ij}}.

We prove the corollary by induction on r: assume every fiber of m_{μ′_{•}} is paved by affines. By the above discussion, \overline{Q}_{|μ′_{•}|} ∩t_{λ}Q_{μ} is a disjoint union of certain (non-empty) locally closed subsets Z_{w}, each of which is isomorphic to an affine space. The boundary of each such Z_{w} is a union of other strata Z_{v}. The triviality statement of Lemma 3.3 and the induction hypothesis then shows that each variety p^{−1}(Z_{w}) is paved by affine spaces.

These remarks imply (by an inductive argument) that m_{μ_{•}}^{−1}(y) is paved by affine spaces. □

Question 3.9. Suppose μ_{•} is a general r-tuple of coweights μ_{i} (not necessarily minuscule). Which fibers m_{μ_{•}}^{−1}(y) are paved by affine spaces? Does every fiber m_{μ_{•}}^{−1}(y) admit a Hessenberg paving, in the sense of [GKM], §1?

4. Equidimensionality results for sums of minuscules

This section concerns what we can say when the μ_{i}’s are not all minuscule. We will consider the fibers of m_{μ_{•}}, where each μ_{i} is a sum of dominant minuscule coweights. Assume y ∈ Q_{λ} ⊂ \overline{Q}_{|μ_{•}|}.

Proposition 4.1. (1) 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⟩.

(2) Suppose each μ′_{i} is a sum of dominant minuscule coweights. Suppose that

m_{μ_{•}}^{−1}(y) ∩ Q_{μ′_{•}} = m_{μ′_{•}}^{−1}(y) ∩ Q_{μ′_{•}}

is non-empty. Then m_{μ_{•}}^{−1}(y) ∩ Q_{μ′_{•}} is equidimensional of dimension

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

Note that Theorem 1.3 follows from part (2), if we take μ′_{•} = μ_{•}.

Proof. Part (2) follows from part (1). Part (1) follows from Theorem 3.1 and the following lemma, whose proof appears in [H] (Proof of Prop. 1.8). □

Lemma 4.2 (The pulling apart lemma). Suppose μ_{i} = ∑_{j} ν_{ij}, for each i, and consider the diagram

\xymatrix{Q̃_{ν_{••}} \ar[r]^{η} & Q̃_{μ_{•}} \ar[r]^{m_{μ•}} & \overline{Q}_{|μ_{•}|},}

where η = m_{ν_{1•}} ˜× ⋯ ˜×m_{ν_{r•}} and hence m_{μ_{•}} ∘ η = m_{ν_{••}}. Then if m_{ν_{••}}^{−1}(e_{λ}) is equidimensional of dimension ⟨ρ, |ν_{••}|−λ⟩, the morphism m_{μ_{•}} satisfies the conclusion of Proposition 4.1, part (1).

Remark 4.3. In general, the fiber m_{μ_{•}}^{−1}(e_{λ}) is not equidimensional of dimension ⟨ρ, |μ_{•}| − λ⟩. Following [KLM], §9.5, consider for example the group G = SO_{5} (so Ĝ = Sp_{4}(ℂ)), where one fundamental weight of Ĝ is minuscule and the other is quasi-minuscule (μ is quasi-minuscule if Ω(V_{μ}) = W μ ∪ {0}). The implication Hecke(μ_{•}, λ) ⇒ Rep(μ_{•}, λ) does not always hold. In fact, let

μ_{1} = μ_{2} = μ_{3} = α_{1} + α_{2} = (1, 1),

where α_{i} are the two simple roots of Ĝ, following the conventions of [Bou]. Let λ = 0. In [KLM] it is shown that V_{μ_{•}}^{λ} = 0 and c_{μ_{•}}^{λ} = q^{5} − q ≠ 0.

We see using Lemma 2.3 that

dim(m_{μ_{•}}^{−1}(e_{0})) = dim(m_{μ_{•}}^{−1}(e_{0}) ∩ Q_{μ_{•}}) = 5

which is strictly less than ⟨ρ, |μ_{•}|⟩ = 6.

Since every coweight of SO_{5} is a sum of minuscule and quasi-minuscule coweights, this example together with Lemma 4.2 yields: if we assume each μ_{i} is minuscule or quasi-minuscule, in general the fibers m_{μ_{•}}^{−1}(y) are not all equidimensional of the maximal possible dimension.

5. Relating structure constants for sums of minuscules

Corollary 5.1. If every μ_{i} is a sum of minuscules, then

Rep(μ_{•}, λ) ⇔ Hecke(μ_{•}, λ).

Proof. The argument is as in [H], which handled the case of GL_{n}. Namely, we prove the implication ⇐ as follows. If Hecke(μ_{•}, λ) holds, then m_{μ_{•}}^{−1}(e_{λ}) ∩ Q_{μ_{•}} ≠ ∅, and then by Proposition 4.1 (2), we see that the dimension of this intersection is ⟨ρ, |μ_{•}| − λ⟩. Hence by Theorem 2.2, the property Rep(μ_{•}, λ) holds. □

Remark 5.2. Note that there is no assumption on the coweight λ. In particular, λ need not be a sum of dominant minuscule coweights. After this result was obtained, an improvement was announced in a preprint of Kapovich-Millson [KM], for the case r = 2. This improvement states that

Hecke(μ_{1}, μ_{2}, λ) ⇒ Rep(μ_{1}, μ_{2}, λ)

as long as at least one of the coweights μ_{1}, μ_{2} or λ is a sum of minuscules (instead of two of them, as required in Corollary 5.1).

6. A new proof of the P-R-V property for sums of minuscules

Before it was established independently by O. Mathieu [Ma] and S. Kumar [Ku], the following was known as the P-R-V conjecture (see also [Li] for a short proof based on Littelmann’s path model).

Theorem 6.1 (P-R-V property). If λ = w_{1}μ_{1} + ⋯ w_{r}μ_{r}, then V_{λ} appears with multiplicity at least 1 in the tensor product V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}}.

It is actually much easier to establish the Hecke-algebra analogue of the P-R-V property.

Proposition 6.2. Under the same assumptions as above, the function f_{λ} appears in f_{μ_{1}} ∗ ⋯ ∗ f_{μ_{r}} with non-zero coefficient.

Proof. Recall that Hecke(μ_{•}, λ) holds if and only if the variety m_{μ_{•}}^{−1}(e_{λ}) ∩ Q_{μ_{•}} is non-empty (see Lemma 2.3 and [H], §4).

But the equality λ = w_{1}μ_{1} +⋯+w_{r}μ_{r} yields a point L_{•} in the intersection m_{μ_{•}}^{−1}(e_{λ})∩Q_{μ_{•}}, given by

L_{i} = t_{w_{1}μ_{1}+···+w_{i}μ_{i}}e_{0},

for 0 ≤ i ≤ r. □

Note that Corollary 5.1 and Proposition 6.2 combine to give a new proof of Theorem 6.1, in the case where each μ_{i} is a sum of minuscule coweights (in particular for the group GL_{n}).

7. A saturation theorem for sums of minuscules

The following saturation property for Hecke(μ_{•}, λ) is due to M. Kapovich, B. Leeb, and J. Millson [KLM].

Theorem 7.1 ([KLM]). For any split group G over 𝔽_{q}, there exists a positive integer k_{G} given explicitly in terms of the root data for G, with the following property: for any tuple of dominant coweights (μ_{•}, λ) satisfying ∑_{i} μ_{i} − λ ∈ Q^{∨}, and every positive integer N, we have

Hecke(Nμ_{•}, Nλ) ⇒ Hecke(k_{G}μ_{•}, k_{G}λ).

We call k_{G} the Hecke algebra saturation factor for G. It turns out that k_{GL_{n}} = 1, so this result shows that the structure constants for the Hecke algebra have the strongest possible saturation property in the case of GL_{n}.

Corollary 5.1 and Theorem 7.1 combine to give the following saturation theorem.

Theorem 7.2 (Saturation for sums of minuscules – weak form). Suppose that at least r − 1 of the weights μ_{i} of Ĝ are sums of dominant minuscule weights, and suppose the sum ∑_{i} μ_{i} belongs to the lattice spanned by the roots of Ĝ. Let N be any positive integer. Then

(V_{Nμ_{1}} ⊗ ⋯ ⊗ V_{Nμ_{r}})Ĝ ≠ 0 ⇒ (V_{k_{G}μ_{1}} ⊗ ⋯ ⊗ V_{k_{G}μ_{r}})Ĝ ≠ 0.

In the case of GL_{n} this was proved in [KLM], providing a new proof of the saturation property for GL_{n}.

Proof. Without loss of generality, we may assume μ_{1}, …, μ_{r−1} are sums of minuscules. Recall that for any highest weight representation V_{μ}, the contragredient (V_{μ})^{∗} is also irreducible, so that we can define a dominant coweight μ^{∗} by the equality (V_{μ})^{∗} = V_{μ^{∗}}. Let μ′_{•} = (μ_{1}, …, μ_{r−1}). Now the theorem follows from Theorem 7.1 and Corollary 5.1, and the equivalences

Hecke(μ_{•}, 0) ⇔ Hecke(μ′_{•}, μ_{r}^{∗})
Rep(μ_{•}, 0) ⇔ Rep(μ′_{•}, μ_{r}^{∗}).
□

In fact it seems that a stronger implication will hold. Although it might not be necessary, here we will assume that all the weights μ_{i} are sums of minuscules, to be consistent with computer checks we ran with LiE. We expect that the saturation factor k_{G} can be omitted in the above statement.

Conjecture 7.3 (Saturation for sums of minuscules – strong form). Suppose each weight μ_{i} of Ĝ is a sum of dominant minuscule weights, and suppose ∑_{i} μ_{i} belongs to the root lattice. Then

(V_{Nμ_{1}} ⊗ ⋯ ⊗ V_{Nμ_{r}})Ĝ ≠ 0 ⇒ (V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}})Ĝ ≠ 0.

When k_{G} is small (e.g. k_{GSp_{2n}} = 2) the conjecture seems to be only a minor strengthening of Theorem 7.2. However for some exceptional groups k_{G} is quite large (e.g. k_{E_{7}} = 12) and there the conjecture indicates that a substantial strengthening of Theorem 7.2 should remain valid. In any case, the conjecture “explains” to a certain extent the phenomenon of saturation for GL_{n} by placing it in a broader context.

We present the following evidence for Conjecture 7.3. Taking Corollary 5.1 into account, the following theorem results immediately from a slightly more comprehensive Hecke-algebra analogue, proved in a joint appendix with M. Kapovich and J. Millson (Theorem 9.7).

Theorem 7.4. Suppose that G_{ad} a product of simple groups of type A, B, C, D, or E_{7}. Suppose each μ_{i} is a sum of minuscules and that ∑_{i} μ_{i} ∈ Q(Ĝ). Then

Rep(Nμ_{•}, 0) ⇒ Rep(μ_{•}, 0),

provided we assume either of the following conditions:

(i) All simple factors of G_{ad} are of type A, B, C, or E_{7}; (ii) All simple factors of G_{ad} are of type A, B, C, D, or E_{7}, and for each factor of type D_{2n}

(resp. D_{2n+1}) the projection of μ_{i} onto that factor is a multiple of a single minuscule weight (resp. a multiple of the minuscule weight ϖ_{1}).

Remark 7.5. M. Kapovich and J. Millson have recently announced in [KM] that the implication

Rep(Nμ_{•}, 0) ⇒ Rep(k_{G}^{2}μ_{•}, 0)

holds for every split semi-simple group G over k((t)) and for all weights μ_{•} (assuming of course ∑_{i} μ_{i} ∈ Q(Ĝ)). Conjecture 7.3 above is in a sense “orthogonal” to this statement: instead of fixing a group and then asking what saturation factor will work for that group, we are asking whether for certain special classes of weights μ_{•} (e.g. sums of minuscules for groups that possess them) the saturation factor of 1 is guaranteed to work.

7.0.1. Relation with the conjecture of Knutson-Tao. The following conjecture of Knutson-Tao proposes a sufficient condition on weights μ_{1}, μ_{2}, λ of a general semi-simple group to ensure a saturation theorem will hold.

Conjecture 7.6 ([KT]). Let Ĝ be a connected semi-simple complex group, and suppose (μ_{1}, μ_{2}, λ) are weights of a maximal torus T̂ such that μ_{1} + μ_{2} + λ annihilates all elements s ∈ T̂ whose centralizer in Ĝ is a semi-simple group. Then for any positive integer N,

(7.0.1)  (V_{Nμ_{1}} ⊗ V_{Nμ_{2}} ⊗ V_{Nλ})Ĝ ≠ 0 ⇒ (V_{μ_{1}} ⊗ V_{μ_{2}} ⊗ V_{λ})Ĝ ≠ 0.

Fix a connected semi-simple complex group Ĝ. It is natural to ask how Conjectures 7.3 and 7.6 are related: if we assume μ_{1}, μ_{2} and λ are sums of minuscules, does the Knutson-Tao conjecture then imply Conjecture 7.3? The answer to this question is no, as the following example demonstrates.

Example. Let Ĝ = Spin(12), the simply-connected group of type D_{6}. Suppose μ_{1}, μ_{2}, λ are three weights of T̂ whose sum belongs to the root lattice (so the sum annihilates the center Z(Ĝ)). Conjecture 7.3 asserts that (7.0.1) holds provided that

(7.0.2)  μ_{1}, μ_{2}, λ ∈ ℤ_{≥0}[ϖ_{1}, ϖ_{5}, ϖ_{6}],

where we have labeled characters using the conventions of [Bou]. Henceforth let us assume condition (7.0.2). Now, Conjecture 7.6 asserts that (7.0.1) holds provided μ_{1} + μ_{2} + λ also annihilates certain elements. Consider the element s := ϖ_{3}^{∨}(e^{2πi/2}) ∈ T̂, an element of order 2. It is easy to check that CentĜ(s) is a semi-simple group. Furthermore, it is clear that μ_{1} + μ_{2} + λ annihilates s if and only if

(7.0.3)  ⟨μ_{1} + μ_{2} + λ, ϖ_{3}^{∨}⟩ ∈ 2ℤ.

But this last condition can easily fail: take for example μ_{1} = ϖ_{6}, μ_{2} = 0, and λ = ϖ_{6}, so that μ_{1} + μ_{2} + λ = 2ϖ_{6} = e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + e_{6} and thus

⟨μ_{1} + μ_{2} + λ, ϖ_{3}^{∨}⟩ = 3.

In other words, if μ_{1}, μ_{2}, λ are sums of minuscules for Spin(12), the Knutson-Tao conjecture predicts at most the implication

(V_{Nμ_{1}} ⊗ V_{Nμ_{2}} ⊗ V_{Nλ})Ĝ ≠ 0 ⇒ (V_{2μ_{1}} ⊗ V_{2μ_{2}} ⊗ V_{2λ})Ĝ ≠ 0,

whereas Conjecture 7.3 predicts the sharper statement (7.0.1). For the example μ_{1} = λ = ϖ_{6}, μ_{2} = 0 above, this sharper statement is indeed correct (use that V_{ϖ_{6}} is a self-contragredient representation).

8. Equidimensionality of (locally closed) partial Springer varieties for GL_{n}

In this section we will use Proposition 4.1 to deduce similar equidimensionality results for “locally closed” Springer varieties associated to a partial Springer resolution of the nilpotent cone for GL_{n}. We will also characterize those which are non-empty and express the number of irreducible components in terms of structure constants. Finally, we describe the relation of these questions with the Springer correspondence. For the most part, our notation closely parallels that of [BM].

8.1. Definitions and the equidimensionality property. Let V denote a k-vector space of dimension n, and let (d_{1}, …, d_{r}) denote an ordered r-tuple of nonnegative integers such that d_{1}+⋯+d_{r} = n. The r-tuple d_{•} determines a standard parabolic subgroup P ⊂ GL(V) = GL_{n}. We consider the variety of partial flags of type P:

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

Consider the Levi decomposition P = LN, where N is the unipotent radical of P, and L ≅ GL_{d_{1}} × ⋯ × GL_{d_{r}}.

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

ξ : Ñ^{P} → N

where N ⊂ End(V) is the nilpotent cone, Ñ^{P} = {(g, V_{•}) ∈ N × P | V_{•} ∈ P_{g}}, and the morphism ξ forgets V_{•}.

The nilpotent cone N has a natural stratification indexed by the partitions of n. These can be identified with dominant coweights λ = (λ_{1}, …, λ_{n}) where λ_{1} ≥ ⋯ ≥ λ_{n} ≥ 0 and λ_{1} + ⋯ + λ_{n} = n. The integers λ_{i} give the sizes of Jordan blocks in the normal form of an element in N. In a similar way, the partial Springer resolution Ñ^{P} carries a natural stratification indexed by r-tuples μ′_{•} = (μ′_{1}, …, μ′_{r}) where μ′_{i} is a partition of d_{i} having length n (see [BM], §2.10). In other words, if we let

μ_{i} = (d_{i}, 0^{n−1})

for 1 ≤ i ≤ r, then Ñ^{P} carries a natural stratification indexed by r-tuples μ′_{•} = (μ′_{1}, …, μ′_{r}) where for each i, μ′_{i} is a dominant coweight for GL_{n} and μ′_{i} ⪯ μ_{i}. The stratum indexed by μ′_{•} consists of pairs (g, V_{•}) such that the Jordan form of the endomorphism on V_{i−1}/V_{i} induced by g has Jordan type μ′_{i}, for 1 ≤ i ≤ r.

Let us denote by N_{λ} ⊂ N the stratum indexed by λ. Write x = μ′_{•} for short and denote by P^{(x)} the stratum of Ñ^{P} which is indexed by x = μ′_{•}.

The morphism ξ : Ñ^{P} → N is a locally-trivial semi-small morphism of stratified spaces. In fact, by the lemma below, it comes by restriction of the morphism m_{μ_{•}} : Q̃_{μ_{•}} → \overline{Q}_{|μ_{•}|} = \overline{Q}_{(n,0^{n−1})}, where μ_{i} = (d_{i}, 0^{n−1}), 1 ≤ i ≤ r.

The following useful relation between the nilpotent cone and the affine Schubert variety \overline{Q}_{(n,0^{n−1})} was observed by Lusztig [Lu1] and Ngô [Ng]. Here, the standard lattice V ⊗_{k}O ≅ O^{n} is the base-point in Q, which we previously denoted by e_{0}.

Lemma 8.1. The morphism g ↦ (g + tI_{n})O^{n} determines an open immersion ι : N ↪ \overline{Q}_{|μ_{•}|}. Furthermore, the restriction of m_{μ_{•}} over N can be identified with ξ. In other words, there is a Cartesian diagram

\xymatrix{Ñ^{P} \ar[r]^{ι̃} \ar[d]_{ξ} & Q̃_{μ_{•}} \ar[d]^{m_{μ•}} \\ N \ar[r]^{ι} & \overline{Q}_{|μ_{•}|}.}

Moreover, for each λ (resp. μ′_{•}), we have ι^{−1}(Q_{λ}) = N_{λ} (resp. ι^{−1}(Q_{μ′_{•}}) = P^{(x)}, where x = μ′_{•}).

Proof. The fact that g ↦ (g + tI_{n})O^{n} determines an open immersion is most easily justified by proving the analogous global statement. We refer to the proof of [Ng], Lemme 2.2.2. for the proof, since this point is not crucial in our applications of this lemma.

The compatibility between m_{μ_{•}} and ξ can be found in that same paper of Ngô (he proves in loc. cit. Lemme 2.3.1 a corresponding global statement). Since this compatibility is used below, we will sketch the proof. It is a direct consquence of the following explicit description of the morphism ι̃.

Suppose (g, V_{•}) ∈ Ñ^{P}. Since deg_{t}(det(g + tI_{n})) = n, the lattice (g + tI_{n})O^{n} has kcodimension n in O^{n}. So, we can identify the k-vector space O^{n}/(g+tI_{n})O^{n} with V, equivariantly for the action of g ∈ End(V) on both sides. The g-stable partial flag V_{•} then determines a g-stable (hence also t-stable) partial flag in O^{n}/(g + tI_{n})O^{n}. Thus, we get a sequence of O-lattices O^{n} = L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{r} = (g + tI_{n})O^{n}, such that dim_{k}(L_{i−1}/L_{i}) = d_{i}, for all 1 ≤ i ≤ r. Hence L_{•} ∈ Q̃_{μ_{•}}, and we have ι̃(g, V_{•}) = L_{•}. □

The goal of this subsection is to prove, in Proposition 8.2 below, an equidimensionality property of the locally closed Springer varieties P_{y}^{(x)}. We define these as follows. Let λ index the stratum N_{λ} of N, and let x = μ′_{•} index the stratum P^{(x)} of Ñ^{P}. Let y ∈ N_{λ}. We define

P_{y}^{(x)} := ξ^{−1}(y) ∩ P^{(x)}.

Put another way,

(8.1.1)  P_{y}^{(x)} = ι^{−1}(m_{μ_{•}}^{−1}(y) ∩ Q_{μ′_{•}}).

Further, let P^{x} = \overline{P^{(x)}} and put P_{y}^{x} = ξ^{−1}(y) ∩ P^{x}. Thus,

(8.1.2)  P_{y}^{x} = ι^{−1}(m_{μ_{•}}^{−1}(y) ∩ \overline{Q}_{μ′_{•}}).

This is essentially the notation used in [BM], §3.2. Following loc. cit., we recall that

• the Steinberg variety P_{y} := ξ^{−1}(y) is the disjoint union of its Springer parts P_{y}^{(x)},

• P_{y} = P_{y}^{x} if x is “regular” (i.e., μ′_{i} = μ_{i}, for all 1 ≤ i ≤ r),

• P_{y}^{(0)} = P_{y}^{0} (where “x = 0” means μ′_{i} = μ′_{i}(0) := (1^{d_{i}}, 0^{n−d_{i}}) for all 1 ≤ i ≤ r).

The varieties P_{y}^{0} are called Spaltenstein varieties in [BM] and Spaltenstein-Springer varieties in [H].

Now Proposition 4.1 and (8.1.1) immediately give us the following equidimensionality result for the locally-closed Springer varieties P_{y}^{(x)}, where x = μ′_{•} and y ∈ N_{λ}. It is quite possible that this result is already known to some experts, but it does not seem to appear in the literature. In any case, the present proof via Proposition 4.1(2) is a very transparent one.

Proposition 8.2. If P_{y}^{(x)} ≠ ∅, then every irreducible component of P_{y}^{(x)} has dimension ⟨ρ, |μ′_{•}| − λ⟩.

In particular, the Spaltenstein-Springer variety P_{y}^{0} (if non-empty), is equidimensional of dimension ⟨ρ, |μ′_{•}(0)| − λ⟩, where μ′_{i}(0) := (1^{d_{i}}, 0^{n−d_{i}}) for each i.

Note that the last statement was proved in [Sp] [final Corollary], by completely different methods. Spaltenstein also proved that the varieties P_{y}^{0} admit pavings by affine spaces, and this fact can now be seen as a special case of Corollary 1.2.

Remark 8.3. Note that we have not proved the equidimensionality of the varieties P_{y}^{x}, and indeed they are not always equidimensional. In fact, it is known that there exist coweights μ_{i} = (d_{i}, 0^{n−1}) and λ ≺ |μ_{•}| = (n, 0^{n−1}) such that the partial Springer fiber P_{y}^{x} is not equidimensional for y ∈ N_{λ}. See [St], proof of Cor. 5.6, or [Sh], Thm. 4.15. By virtue of Lemma 8.1, the corresponding fibers m_{μ_{•}}^{−1}(y) are also not equidimensional.

8.2. When are locally closed Springer varieties non-empty? Let x = μ′_{•} and y ∈ N_{λ}. It is clear that P_{y}^{x} ≠ ∅ if and only if λ ⪯ |μ′_{•}|. The non-emptiness of the locally closed varieties P_{y}^{(x)} is more subtle.

Proposition 8.4. The locally-closed Springer variety P_{y}^{(x)} is non-empty if and only if Rep(μ′_{•}, λ) holds. Furthermore, there are equalities

# irreducible components of P_{y}^{(x)} = # top-dimensional irreducible components of P_{y}^{x}

= # top-dimensional irreducible components of m_{μ′_{•}}^{−1}(y)

= dim(V_{μ′_{•}}^{λ}).

This is obvious from (8.1.1) and our previous discussion.

8.3. Relation with the Springer correspondence. The question of when P_{y}^{(x)} ≠ ∅ can also be related to the Springer correspondence. For this discussion we assume k = ℂ and temporarily replace GL_{n} with any connected reductive group G. The Springer correspondence is a cohomological realization of a one-to-one correspondence

ρ(y, ψ) ↔ (y, ψ)

between irreducible representations ρ of W and the set of relevant pairs (y, ψ), where y is a stratum of N and ψ is a representation of the fundamental group of that stratum, giving rise to a local system L_{ψ}. See [BM], Theorem 2.2.

Let V_{(y,ψ)} denote the underlying vector space for the representation ρ(y, ψ). Then the Weyl group W acts on the cohomology of the Steinberg variety

H^{i}(B_{y}, ℚ),

and in fact if we let d_{y} := dim(B_{y}), we have the isomorphism of W -modules

H^{2d_{y}}(B_{y}, ℚ)^{ρ(y,1)} = V_{(y,1)},

where the left-hand side denotes the isotypical component of type ρ(y, 1). See [BM], §2.2.

Now once again we assume G = GL_{n} (for the rest of this section). In this case, it is known that only the representations ρ(y, 1) arise, and they give a complete list of the irreducible representations of W = S_{n}.

In the sequel, the symbol y will either denote a point y ∈ N_{λ}, or the stratum y = λ itself. Similarly, sometimes x will denote a point x ∈ Q_{μ′_{•}}, and other times it will denote the stratum x = μ′_{•} itself. Hopefully context will make it clear what is meant in each case. Note that d_{y} = dim(B_{y}) = ⟨ρ, |μ_{•}| − λ⟩ in this case.

Let W (L) = N_{L}(T)/T denote the Weyl group of the standard Levi subgroup L of P we already fixed. Let B(L) (resp. N (L) ) denote the flag variety (resp. nilpotent cone) for L, and for ℓ ∈ N (L), let B(L)_{ℓ} denote the corresponding Steinberg variety. As in [BM], §2.10, we can regard any index x = μ′_{•} as corresponding to a unique nilpotent orbit ℓ: the choice of x and ℓ both amount to choosing an r-tuple (μ′_{1}, ⋯, μ′_{r}) where μ′_{i} is a partition of d_{i} of length n. Thus, we can also write B(L)_{ℓ} = B(L)_{x}.

The question of whether P_{y}^{(x)} ≠ ∅ is essentially equivalent to whether ρ(x, 1) appears in the restriction to W (L) of the W -module H^{2d_{y}}(B_{y}, ℚ).

Proposition 8.5. The locally-closed Springer variety P_{y}^{(x)} is non-empty if and only if the representation ρ(x, 1) of W (L) appears with positive multiplicity in H^{2d_{y}}(B_{y}, ℚ)|_{W (L)}. Furthermore, the multiplicity is given by the formula

dim_{ℂ} [Hom_{W (L)}(ρ(x, 1), H^{2d_{y}}(B_{y}, ℚ)|_{W (L)})] = # top-dimensional irreducible components of P_{y}^{x}
= dim(V_{μ′_{•}}^{λ}).

Proof. Since V_{(x,1)} = H^{2d_{x}}(B(L)_{x}, ℚ)^{ρ(x,1)}, the first statement will follow from the proof of [BM], Theorem 3.3, which shows in effect that there is an isomorphism of W (L)-modules

H^{2d_{y}−2d_{x}}(P_{y}^{x}, IC(P^{x})) ⊗ H^{2d_{x}}(B(L)_{x}, ℚ)^{ρ(x,1)} = [H^{2d_{y}}(B_{y}, ℚ)|_{W (L)}]^{ρ(x,1)}.

Here P^{x} := \overline{P^{(x)}} and IC(P^{x}) denotes the intersection complex of P^{x}, following the conventions of loc. cit. (it is a complex supported in cohomological degrees [0, dim(P^{x}))).

Now we note that, provided P_{y}^{(x)} ≠ ∅,

dim(P_{y}^{x}) = ⟨ρ, |μ′_{•}| − λ⟩ = ⟨ρ, |μ_{•}| − λ⟩ − ⟨ρ, |μ_{•}| − |μ′_{•}|⟩ = d_{y} − d_{x}.

Here, we have used the isomorphism

B(L)_{x} ≅ η^{−1}(x)

of [BM], Lemma 2.10 (b) to justify the equality d_{x} = ⟨ρ, |μ_{•}| − |μ′_{•}|⟩. Finally, it is well-known that since ξ is semi-small, the dimension of H^{2d_{y}−2d_{x}}(P_{y}^{x}, IC(P^{x})) is the number of irreducible components of P_{y}^{x} having dimension d_{y} − d_{x} (see e.g. [H], Lemma 3.2). By Proposition 8.4, we are done.

□

This gives a refinement and new proof of [BM] Corollary 3.5, in the case of GL_{n}.

9. Appendix: constructing special r-gons in Bruhat-Tits buildings by reduction to rank 1

by Thomas J. Haines and Michael Kapovich and John J. Millson 9.1. Constructing r-gons with allowed side-lengths. Let G denote a connected reductive group over an algebraically closed field k. Let O = k[[t]] and L = k((t)). (We use the symbol L in place of F to emphasize that k can be any algebraically closed field, and not just \overline{𝔽}_{p} as in the main body of the paper.) Further, let K = G(O) and define the affine Grassmannian Q = G(L)/K, viewed as an ind-scheme over k. We fix once and for all a maximal torus T ⊂ G and a Borel subgroup B = T U containing T.

For a cocharacter μ of T, we shall denote by \overline{μ} the cocharacter of the adjoint group G_{ad} which results by composing μ with the homomorphism G → G_{ad}. Recall that G_{ad} is a product of simple adjoint groups H, and we will denote by \overline{μ}_{H} the composition of \overline{μ} with the projection G_{ad} → H. We have \overline{μ}_{H} ∈ X_{∗}(T_{H}), where T_{H} is the image of the torus T under the homomorphism G → H.

Throughout this appendix, dominant coweight means B-dominant cocharacter. Similar terminology will apply to the quotients H (we use the Borel B_{H} which is the image of B).

Recall that each factor H corresponds to an irreducible finite root system whose Weyl group possesses a unique longest element w_{H,0}. For any coweight ν of T_{H}, we set ν^{∗} = −w_{H,0}ν. We call such a coweight ν self-dual if ν^{∗} = ν.

Let Q^{∨}(H) (resp. P^{∨}(H) = X_{∗}(T_{H})) denote the coroot (resp. coweight) lattice of the adjoint group H.

Our first result is the following generalization of Proposition 7.7 of [KM]. To state it, we need to single out a special class of fundamental coweights.

Definition 9.1. Let ϖ_{i}^{∨} denote a a fundamental coweight of an adjoint group H. We call ϖ_{i}^{∨} allowed if it satisfies the following properties:

(1) ϖ_{i}^{∨} is self-dual;

(2) nϖ_{i}^{∨} ∈ Q^{∨}(H) ⇔ n ∈ 2 ℤ.

Proposition 9.2. Suppose that for each factor H of G_{ad} we are given an allowed fundamental coweight λ_{H} ∈ X_{∗}(T_{H}).

Suppose that for each i = 1, 2, …, r, the image \overline{μ_{i}} of the dominant coweight μ_{i} ∈ X_{∗}(T) is a sum of the form

\overline{μ_{i}} = ∑_{H} a_{i}^{H}λ_{H}

for nonnegative integers a_{i}^{H}. Suppose that ∑_{i} μ_{i} ∈ Q^{∨}(G) and that the coweights μ_{i} satisfy the following weak generalized triangle inequalities

(9.1.1)  μ_{i}^{∗} ⪯ μ_{1} + ⋯ + μ̂_{i} + ⋯ + μ_{r}.

Then the variety Q_{μ_{•}} ∩ m_{μ_{•}}^{−1}(e_{0}) is non-empty.

In the terminology of [KLM], [KM], the building of G(L) has a closed r-gon with sidelengths μ_{1}, …, μ_{r}, whose vertices are special vertices. We call these special r-gons.

Proof. For each factor H, let α_{H} denote the simple B-positive root corresponding to the fundamental coweight λ_{H}. We consider the Levi subgroup M ⊂ G that is generated by T along with the root groups for all the roots ±α_{H}:

M := ⟨T, U_{α_{H}} , U_{−α_{H}} ⟩.

The coweights μ_{i} resp. λ_{H} determine coweights for M resp. M_{ad}; we write \overline{\overline{μ_{i}}} resp. \overline{\overline{λ_{H}}} for their images in the adjoint group M_{ad}. Note that

M_{ad} = ∏_{H} PGL_{2},

and that in the factor indexed by H, we can identify \overline{\overline{α_{H}}} = e_{1} − e_{2} and \overline{\overline{λ_{H}}} = (1, 0).

Now our assumptions imply that for each factor H, • a_{1}^{H} + ⋯ + a_{r}^{H} is even, and

• a_{i}^{H} ≤ a_{1}^{H} + ⋯ + â_{i}^{H} + ⋯ + a_{r}^{H}, for each i. As we shall see in the next lemma, these properties imply that there is a special r-gon in

the building for PGL_{2} with side-lengths a_{1}^{H}, …, a_{r}^{H}. Note that since k is infinite, we will be working with a tree having infinite valence at each vertex, but this causes no problems.

Lemma 9.3. Suppose u_{1}, …, u_{r} are nonnegative integers satisfying the generalized triangle inequalities

u_{i} ≤ u_{1} + ⋯ + û_{i} + ⋯ + u_{r}, ∀i,
∑_{i} u_{i} ≡ 0 (mod 2).

Then there exists a special r-gon in the tree B(PGL_{2}) having side lengths u_{1}, …, u_{r}.

Proof. First we claim that there exist integers l and m, with 1 ≤ l < m ≤ r, such that if we set

A = u_{1} + ⋯ + u_{l}
B = u_{l+1} + ⋯ + u_{m}
C = u_{m+1} + ⋯ + u_{r},

then

A ≤ B + C
B ≤ A + C
C ≤ A + B
A + B + C ≡ 0 (mod 2).

Indeed, note that for all i, u_{i} ≤ \frac{1}{2}(∑_{i} u_{i}). We may choose l to be the largest such that

u_{1} + ⋯ + u_{l} ≤ \frac{1}{2}(∑_{i} u_{i}),

and then set m = l + 1 (note that necessarily l ≤ r − 1, if at least one u_{i} > 0).

Now given A, B, C as above, we may construct a “tripod” in the building as follows. Choose any vertex v_{0}, and construct a tripod, centered at v_{0}, with legs having lengths l_{1}, l_{2}, l_{3}, where

l_{1} = \frac{A + B − C}{2}
l_{2} = \frac{B + C − A}{2}
l_{3} = \frac{A + C − B}{2} .

Let v_{1}, v_{2}, v_{3} denote the extreme points of the tripod, and consider the oriented paths [v_{0}, v_{i}], where we have labeled vertices in such a way the length of [v_{0}, v_{i}] is l_{i}.

This yields a special 3-gon: the three “sides” are

[v_{3}, v_{0}] ∪ [v_{0}, v_{1}], (length = A)
[v_{1}, v_{0}] ∪ [v_{0}, v_{2}], (length = B)
[v_{2}, v_{0}] ∪ [v_{0}, v_{3}], (length = C).

This (oriented) triangle begins and ends at the special vertex v_{3}. The sides are themselves partitioned into smaller intervals of lengths u_{1}, u_{2}, …, u_{l}, etc. Thus we have a special r-gon with the desired side-lengths. □

The building for M_{ad} is simply the direct product of the buildings for the various PGL_{2} factors, hence we have a special r-gon in the building for M_{ad} with side lengths \overline{\overline{μ}}_{1}, …, \overline{\overline{μ}}_{r}. Equivalently, we have

1_{M_{ad}} ∈ M_{ad}(O)\overline{\overline{μ}}_{1}M_{ad}(O) ⋯ M_{ad}(O)\overline{\overline{μ}}_{r}M_{ad}(O).

Now we want to claim that this implies that

1_{M} ∈ M(O)μ_{1}M(O) ⋯ M(O)μ_{r}M(O).

But this follows from the Lemma 9.4 below. Since M(O) ⊂ K, this immediately implies that

1_{G} ∈ Kμ_{1}K ⋯ Kμ_{r}K,

and thus Q_{μ_{•}} ∩ m_{μ_{•}}^{−1}(e_{0}) ≠ ∅, as desired. □

9.2. Enumerating allowed coweights in H. Proposition 9.2 is most interesting for groups which are not of type A. For each type of adjoint simple factor H not of type A, we enumerate the allowed and the minuscule fundamental coweights. We follow the indexing conventions of [Bou] (note that our coweights are weights for the dual root system).

Type of H  Allowed fundamental coweights  Minuscule coweights
B_{n}  ϖ_{i}^{∨}, i odd  ϖ_{1}^{∨}
C_{n}  ϖ_{n}^{∨}  ϖ_{n}^{∨}
D_{2n}  ϖ_{2i−1}^{∨} (1 ≤ i ≤ n − 1), ϖ_{2n−1}^{∨}, ϖ_{2n}^{∨} ϖ_{1}^{∨}, ϖ_{2n−1}^{∨}, ϖ_{2n}^{∨}
D_{2n+1}  ϖ_{2i−1}^{∨}, 1 ≤ i ≤ n  ϖ_{1}^{∨}, ϖ_{2n}^{∨}, ϖ_{2n+1}^{∨}
E_{6}  -  ϖ_{1}^{∨}, ϖ_{6}^{∨}
E_{7}  ϖ_{2}^{∨}, ϖ_{5}^{∨}, ϖ_{7}^{∨}  ϖ_{7}^{∨}
E_{8}  -  -
F_{4}  -  -
G_{2}  -  -

Note that for each case where H possesses a unique minuscule coweight (B_{n}, C_{n}, E_{7}), that minuscule coweight is allowed. For type D_{2n}, all three minuscule coweights are allowed, but for D_{2n+1}, only ϖ_{1}^{∨} is allowed.

9.3. Reduction to adjoint groups.

Lemma 9.4. For any connected reductive algebraic group G over k and dominant coweights μ_{i} such that ∑_{i} μ_{i} ∈ Q^{∨}(G), we have the following two statements.

(1) The canonical homomorphism π : G(O) → G_{ad}(O) is surjective.

(2) Let Z denote the center of G. If z ∈ Z(L) satisfies

z1_{G} ∈ G(O)μ_{1}G(O) ⋯ G(O)μ_{r}G(O)

then z ∈ Z(O) ⊂ G(O), and thus the statement holds with z omitted.

Proof. Property (1) is a standard fact resulting from Hensel’s lemma (see e.g. [PR], Lemma 6.5). Let us recall briefly the proof. For a ∈ G_{ad}(O), the preimage π^{−1}(a) in G(O) is the set of O-points of a smooth O-scheme (a torsor for the smooth O-group scheme Z_{O}). Clearly the reduction modulo t of π^{−1}(a) has a k-point (the residue field O/(t) = k being assumed algebraically closed). Now by Hensel’s lemma, π^{−1}(a) also has an O-point, proving property (1).

For Property (2) let us consider first the case of GL_{n}. The hypothesis implies that zI_{n} belongs to the kernel of the homomorphism

val ∘ det : GL_{n}(L) → ℤ,

since both K and Q^{∨}(GL_{n}) ↪ T (L) belong to that kernel. But then it is clear that z ∈ O^{×}.

In general, the same argument works if we replace val ∘ det with the Kottwitz homomorphism

ω_{G} : G(L) → X^{∗}(Z(Ĝ))_{I},

where I denotes the inertia group Gal(L^{sep}/L) ^{2}; see [Ko], §7 for the construction and properties of this map (we will use the functoriality of G ↦ ω_{G} below). Indeed, since G(O) and Q^{∨}(G) ↪ T (L) belong to the kernel of ω_{G}, so does z. Therefore, we will be done once we justify the equality

Z(L) ∩ ker(ω_{G}) = Z(O).

Suppose z ∈ Z(L) ∩ ker(ω_{G}). Let T denote a (split) maximal k-torus of G, and consider the composition

\xymatrix{Z \ar[r] & T \ar[r]^{c} & D := G/G_{der}.}

By the functoriality of ω_{G}, z ∈ ker(ω_{G}) implies that c(z) ∈ ker(ω_{D}). Since D_{L} is a split torus over L, the latter kernel is D(O). Now since

Z(O) → D(O)

is surjective (by the same proof as in part (1)), there exists z_{0} ∈ Z(O) such that

z_{0}^{−1}z ∈ ker[G(L) → D(L)] = G_{der}(L).

But Z(L) ∩ G_{der}(L) = Z(k) ∩ G_{der}(k) (since the latter is a finite group), which obviously belongs to Z(O). This implies that z ∈ Z(O), as claimed. □

As a corollary of the proof, we have

Corollary 9.5. For any tuple (μ_{•}, λ) such that ∑_{i} μ_{i} − λ ∈ Q^{∨}(G),

Hecke^{G}(μ_{•}, λ) ⇔ Hecke^{G_{ad}}(\overline{μ}_{•}, \overline{λ}).

The dual of the homomorphism

T → T_{ad}

is the composition

T̂_{sc} ↠ T̂_{der} ↪ T̂,

where T̂_{der} := T̂∩Ĝ_{der} and T̂_{sc} is the preimage of T̂_{der} under the isogeny Ĝ_{sc} → Ĝ_{der}. Viewing a coweight λ ∈ X_{∗}(T) as a weight for the dual torus T̂, we let \overline{λ} denote its image under the map

X^{∗}(T̂) → X^{∗}(T̂_{sc}).

With this notation, Corollary 9.5 has the following analogue.

2Since G is split over L, we may omit the coinvariants under I here.

Lemma 9.6. For any tuple (μ_{•}, λ) of weights such that ∑_{i} μ_{i} − λ ∈ Q(Ĝ),

RepĜ(μ_{•}, λ) ⇔ RepĜ_{sc}(\overline{μ}_{•}, \overline{λ}).

Proof. Use the fact that the restriction of V_{λ} ∈ Rep(Ĝ) along Ĝ_{sc} → Ĝ is simply V\overline{λ} ∈ Rep(Ĝ_{sc}). □

9.4. A saturation theorem for Hecke algebra structure constants. Assume now that λ = 0 and ∑_{i} μ_{i} ∈ Q^{∨}(G).

Theorem 9.7. Suppose that G_{ad} a product of simple groups of type A, B, C, D, or E_{7}. Suppose that the projection of each μ_{i} onto a simple adjoint factor of G_{ad} having type B, C, D or E_{7} is a multiple of a single allowed coweight. Then

Hecke(Nμ_{•}, 0) ⇒ Hecke(μ_{•}, 0).

In particular, this conclusion holds if each μ_{i} is a sum of minuscule coweights, provided we assume either of the following conditions:

(i) All simple factors of G_{ad} are of type A, B, C, or E_{7}; (ii) All simple factors of G_{ad} are of type A, B, C, D, or E_{7}, and for each factor of type D_{2n}

(resp. D_{2n+1}) the projection of μ_{i} onto that factor is a multiple of a single minuscule coweight (resp. a multiple of the minuscule coweight ϖ_{1}^{∨}).

Proof. By Corollary 9.5 we can assume G is adjoint, and then prove the saturation property one factor at a time. For factors of type A, the desired saturation property follows from [KLM], Theorem 1.8. For factors of type B, C, D or E_{7}, observe that the assumption Hecke(Nμ_{•}, 0) implies that the weak generalized triangle inequalities (9.1.1) hold, and then use Proposition 9.2. □

When each μ_{i} is a sum of minuscules, it is very probable that the implication holds with no assumption on G_{ad}, in other words, factors D and E_{6} should be allowed in (i) (see Conjecture 7.3). There is ample computer evidence corroborating this. However, the method of reduction to rank 1 used above breaks down for type E_{6} and yields only limited information for type D, and thus a new idea seems to be required.

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