Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra

Thomas J. Haines, Alexandra Pettet

Abstract We consider the “antidominant” variants Θ_{λ}^{−} of the elements Θ_{λ} occurring in the Bernstein presentation of an affine Hecke algebra H. We find explicit formulae for Θ_{λ}^{−} in terms of the Iwahori-Matsumoto generators T_{w} (w ranging over the extended affine Weyl group of the root system R), in the case (i) R is arbitrary and λ is a minuscule coweight, or (ii) R is attached to GL_{n} and λ = me_{k}, where e_{k} is a standard basis vector and m ≥ 1.

In the above cases, certain minimal expressions for Θ_{λ}^{−} play a crucial role. Such minimal expressions exist in fact for any coweight λ for GL_{n}. We give a sheaf-theoretic interpretation of the existence of a minimal expression for Θ_{λ}^{−}: the corresponding perverse sheaf on the affine Schubert variety X(t_{λ}) is the push-forward of an explicit perverse sheaf on the Demazure resolution m : X̃(t_{λ}) → X(t_{λ}). This approach yields, for a minuscule coweight λ of any R, or for an arbitrary coweight λ of GL_{n}, a conceptual albeit less explicit expression for the coefficient Θ_{λ}^{−}(w) of the basis element T_{w}, in terms of the cohomology of a fiber of the Demazure resolution.

AMS Subject classification: 20C08, 14M15.

1 Introduction

Let H be the affine Hecke algebra associated to a root system. There are two well-known presentations of this algebra by generators and relations, the first discovered by Iwahori-Matsumoto [7] and the second by Bernstein [12], [14]; cf. 2.2.1 below. The Iwahori-Matsumoto presentation reflects the structure of the Iwahori-Hecke algebra C_{c}^{∞}(I\G/I) of the split p-adic group G attached to the root system: the generators T_{w} correspond to the characteristic functions of Iwahori double cosets IwI, where w ranges over the extended affine Weyl group. The Bernstein presentation reflects the description of the Hecke algebra as an equivariant K-theory of the associated Steinberg variety, which plays a role in the classification of the representations of H, see [9], [2]. The Bernstein presentation has the advantage that one can construct a basis for the center of H by summing the generators Θ_{λ} over Weyl-orbits of coweights λ; the resulting functions are known as Bernstein functions.

It is of interest to give an explicit relation between the generators in these two presentations. More precisely, one would like to write each Θ_{λ} as an explicit linear combination of the Iwahori-Matsumoto basis elements T_{w}. A direct consequence would be the explicit description of the Bernstein functions (and thus the center of H) in terms of the Iwahori-Matsumoto basis. This problem was considered earlier by the first author [4],[5] because of certain applications to the study of Shimura varieties, and was completely answered there for the case where λ is a minuscule coweight. More recently, O. Schiffmann [16] has given explicit formulae for all elements in a certain basis for the center Z(H) of an affine Hecke algebra H of type A; from this one can derive a formula for the Bernstein function z_{μ}, where μ is any dominant coweight of a group of type A.

In this paper we consider the “antidominant” variants Θ_{λ}^{−} of the elements Θ_{λ}. The support of these functions is somewhat more regular than the original functions Θ_{λ}, cf. Lemma 2.1. In section 3 we consider the case where λ is a minuscule coweight, and we prove the following explicit formula for Θ_{λ}^{−}.

Theorem 1.1 Let λ ∈ X_{∗} be minuscule. Then

Θ_{λ}^{−} = ∑_{{x: λ(x)=λ}} R̃_{x,t_{λ}}(Q)T̃_{x}.

Here λ(x) is the translation part of x “on the left” defined by the decomposition x = t_{λ(x)}w (w ∈ W_{0}), T̃_{x} is a renormalization of the usual Iwahori-Matsumoto generator T_{x}, Q = q^{−1/2} − q^{1/2}, and R̃_{x,y}(Q) is a variant of the usual R-polynomial of Kazhdan-Lusztig [8].

The formula above is analogous to the expression for Θ_{λ} found by the first author in Proposition 4.4 of [5]:

Θ_{λ} = ∑_{{x: t(x)=λ}} R̃_{x,t_{λ}}(Q)T̃_{x}.

Here t(x) is the translation part of x “on the right” defined by the decomposition x = wt_{t(x)} (w ∈ W_{0}). However our proof is simpler and more direct than that of loc.cit., and the same arguments appearing here also give a short proof of the formula for Θ_{λ}. In fact one can derive the formula for Θ_{λ}^{−} from that for Θ_{−λ}, and vice-versa. Indeed, if ι : H → H denotes the anti-involution determined by q^{1/2} ↦ q^{1/2} and T_{x} ↦ T_{x^{−1}}, then ι(Θ_{−λ}) = Θ_{λ}^{−} and ι interchanges the formulae for Θ_{−λ} and Θ_{λ}^{−}. We remark that Theorem 1.1 remains valid for Hecke algebras with arbitrary parameters.

In the fourth section we study coweights of GL_{n} of the form λ = me_{k}, where e_{k} is the k-th standard basis vector and m ∈ ℤ_{+}. The case m = 1, studied in [4] and [5], has relevance to a certain family of Shimura varieties with bad reduction, known as the Drinfeld case. The general case is referred to as multiples of the Drinfeld case. We prove the following formula for Θ_{me_{k}}^{−}.

Theorem 1.2 Let 1 ≤ k ≤ n and m ≥ 1. Then

Θ_{me_{k}}^{−} = ∑_{{x: λ(x)⪯me_{k}}} R̃_{x,t_{me_{k}}} (Q)T̃_{x}.

Here ⪯ denotes the usual partial order on the lattice X_{∗}.

Theorems 1.1 and 1.2 yield explicit expressions for the Bernstein functions z_{μ} (μ minuscule) and z_{me_{1}}, respectively; see Corollary 3.6 and 4.2. The expressions in these special cases seem much simpler than the corresponding ones given by Schiffmann [16].

Theorems 1.1 and 1.2 rely on the existence of certain minimal expressions for Θ_{λ}^{−}: these are expressions of the form

Θ_{λ}^{−} = T̃_{t_{1}}^{ϵ_{1}} ⋯ T̃_{t_{r}}^{ϵ_{r}} T̃_{τ} ,

where t_{λ} = t_{1} ⋯ t_{r}τ (t_{i} ∈ S_{a}, τ ∈ Ω) is a reduced expression and ϵ_{i} ∈ {1, −1} for every 1 ≤ i ≤ r. In the final two sections, we discuss how one can approach a general formula for Θ_{λ}^{−} when λ is an arbitrary coweight of GL_{n}, through minimal expressions (which always exist in this setting, cf. section 5). The result is much less explicit than Theorems 1.1 and 1.2, and involves the geometry of the Demazure resolution X̃(t_{λ}) → X(t_{λ}) of the affine Schubert variety X(t_{λ}). We define a perverse sheaf Ξ_{λ}^{−} on the affine flag variety whose corresponding function in the Hecke algebra is ε_{λ}Θ_{λ}^{−}. It turns out that Ξ_{λ}^{−} is supported on X(t_{λ}). We see that the existence of a minimal expression for Θ_{λ}^{−} is analogous to the existence of a certain explicitly determined perverse sheaf on X̃(t_{λ}) whose push-forward to X(t_{λ}) is Ξ_{λ}^{−}. More precisely, we conclude the paper with the following result (cf. Theorem 6.7, Corollary 6.8 for a completely precise statement).

Theorem 1.3 Let λ be a minuscule coweight of a root system, or any coweight for GL_{n}. Choose a minimal expression for Θ_{λ}^{−} and let m : X̃(t_{λ}) → X(t_{λ}) denote the corresponding Demazure resolution for X(t_{λ}). Then there exists an explicit perverse sheaf D on X̃(t_{λ}) (determined by the choice of minimal expression for Θ_{λ}^{−}) such that

Rm_{∗}(D) = Ξ_{λ}^{−}.

Consequently, if we denote the coefficient of T_{x} in the expression for Θ_{λ}^{−} by Θ_{λ}^{−}(x), then we have

Θ_{λ}^{−}(x) = ε_{λ}Tr(Fr_{q}, H^{•}(m^{−1}(x), D)),

for any x ≤ t_{λ} in the Bruhat order. Here the right hand side denotes the alternating trace of Frobenius on the étale cohomology of the fiber over x ∈ X(t_{λ}) with coefficients in the sheaf D.

Given a coweight λ for an arbitrary root system, let λ_{d} denote the dominant coweight in its Weyl-orbit. We remark that there is a similar formula for Θ_{λ}^{−}, provided that λ_{d} is a sum of minuscule dominant coweights.

2 Preliminaries

2.1 Affine Weyl group

Let (X^{∗}, X_{∗}, R, Ř, Π) be a root system, where Π is the set of simple roots. The Weyl group W_{0} is generated by the set of simple reflections {s_{α} : α ∈ Π}.

We define a partial order ⪯ on X_{∗} (resp. X^{∗}) by setting λ ⪯ μ whenever μ − λ is a linear combination with nonnegative integer coefficients of elements of {α̌ : α ∈ Π} (resp. {α : α ∈ Π}). We let Π_{m} denote the set of roots β ∈ R such that β is a minimal element of R ⊂ X^{∗} with respect to ⪯.

In section 4 we will use the following description of the relation ⪯ for coweights of GL_{n}: (λ_{1}, …, λ_{n}) ⪯ (μ_{1}, … μ_{n}) if and only if λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i} for 1 ≤ i ≤ n − 1, and λ_{1} + ⋯ + λ_{n} = μ_{1} + ⋯ + μ_{n}.

Let W̃ be the semidirect product X_{∗} ⋊ W_{0} = {t_{x}w : w ∈ W_{0}, x ∈ X_{∗}}, with multiplication given by t_{x}wt_{x′}w′ = t_{x+w(x′)}ww′. For any x ∈ W̃, there exists a unique expression t_{λ(x)}w, where w ∈ W_{0} and λ(x) ∈ X_{∗}.

Let

S_{a} = {s_{α} : α ∈ Π} ∪ {t_{−α̌}s_{α} : α ∈ Π_{m}} ⊂ W̃.

Define length l : W̃ → ℤ by

l(t_{x}w) = ∑_{α∈R^{+}:w^{−1}(α)∈R^{−}} |⟨α, x⟩ − 1| + ∑_{α∈R^{+}:w^{−1}(α)∈R^{+}} |⟨α, x⟩|.

Let Q̌ be the subgroup of X_{∗} generated by Ř. The subgroup W_{a} = Q̌ ⋊ W_{0} of W̃ is a Coxeter group with S_{a} the set of simple reflections. The subgroup is normal and admits a complement Ω = {w ∈ W̃ : l(w) = 0}.

For w ∈ W̃ denote ε_{w} = (−1)^{l(w)} and q_{w} = q^{l(w)} (for q any parameter).

The Coxeter group (W_{a}, S_{a}) comes equipped with the Bruhat order ≤. We extend it to W̃ as follows: we say wτ ≤ w′τ′ (w, w′ ∈ W_{a}, τ, τ′ ∈ Ω) if w ≤ w′ and τ = τ′.

Let μ ∈ X_{∗} be dominant. Following Kottwitz-Rapoport [10], we say x ∈ W̃ is μ-admissible if x ≤ t_{w(μ)} for some w ∈ W_{0}. We denote the set of μ-admissible elements by Adm(μ).

2.2 Hecke algebra

2.2.1 Presentations

The braid group of W̃ is the group generated by T_{w} (w ∈ W̃) with relations

T_{w}T_{w′} = T_{ww′} whenever l(ww′) = l(w) + l(w′).

The Hecke algebra H is defined to be the quotient of the group algebra (over ℤ[q^{1/2}, q^{−1/2}]) of the braid group of W̃, by the two-sided ideal generated by the elements

(T_{s} + 1)(T_{s} − q),

for s ∈ S_{a}. The image of T_{w} in H is again denoted by T_{w}. It is known that the elements T_{w} (w ∈ W̃) form a ℤ[q^{1/2}, q^{−1/2}]-basis for H. The presentation of H using the generators T_{w} and the above relations is called the Iwahori-Matsumoto presentation.

For any T_{w}, define a renormalization T̃_{w} = q^{−l(w)/2}T_{w}. Define an indeterminate Q = q^{−1/2} − q^{1/2}. The elements T̃_{w} form a basis for H, and the usual relations can be written as

T̃_{s}T̃_{w} = \{ \begin{matrix}T̃_{sw}, & if l(sw) = l(w) + 1, \\ −QT̃_{w} + T̃_{sw}, & if l(sw) = l(w) − 1, \end{matrix}

for w ∈ W̃ and s ∈ S_{a}. There is also a right-handed version of this relation. Note that T̃_{s}^{−1} = T̃_{s} + Q. We will denote T̃_{t_{λ}} (λ ∈ X_{∗}) simply by T̃_{λ}. For λ ∈ X_{∗}, define

Θ_{λ} = T̃_{λ_{1}} T̃_{λ_{2}}^{−1}

where λ = λ_{1} − λ_{2}, and λ_{1}, λ_{2} are dominant. The elements Θ_{λ} generate a commutative subalgebra of H. It is known that the elements Θ_{λ}T_{w} (λ ∈ X_{∗}, w ∈ W_{0}) form a ℤ[q^{1/2}, q^{−1/2}]-basis for H. These generators satisfy well-known relations (see Prop. 3.6, [14]); in case the root system is simply connected, these are given by the formula

Θ_{λ}T_{s} − T_{s}Θ_{s(λ)} = (q − 1)\frac{Θ_{λ} − Θ_{s(λ)}}{1 − Θ_{−α̌}} ,

where s = s_{α} and α ∈ Π. The presentation of H with generators Θ_{λ}T_{w} and the above relations is called the Bernstein presentation.

We also define

Θ_{λ}^{−} = T̃_{λ′_{1}} T̃_{λ′_{2}}^{−1}

where λ = λ′_{1} − λ′_{2}, and λ′_{1}, λ′_{2} are antidominant.

The involution a → \overline{a} of ℤ[q^{1/2}, q^{−1/2}] determined by q ↦ q^{−1} extends to an involution h → \overline{h}, given by

\overline{∑ a_{w}T_{w}} = ∑ ā_{w}T_{w^{−1}}^{−1}.

It is immediate that \overline{Θ}_{λ} = Θ_{λ}^{−}. Clearly the Bernstein presentation gives rise to an analogous presentation using the generators Θ_{λ}^{−}T_{w} in place of Θ_{λ}T_{w}.

2.2.2 Bernstein functions

For each W_{0}-orbit M in X_{∗}, define the Bernstein function z_{M} attached to M by

z_{M} = ∑_{λ∈M} Θ_{λ}.

When the W_{0}-orbit M contains the dominant element μ, this function is denoted by z_{μ}.

From Corollary 8.8 of Lusztig [12], we have z_{μ} = \overline{z}_{μ}. Consequently,

z_{μ} = ∑_{λ∈W_{0}(μ)} Θ_{λ}^{−}.

2.2.3 A support property

The preceding formula implies that when one studies Bernstein functions there is no harm in working with the functions Θ_{λ}^{−} instead of the functions Θ_{λ}. We do so in this paper because their supports enjoy a nice regularity property, given by the following lemma.

Lemma 2.1 For λ ∈ X_{∗}, we have

supp(Θ_{λ}^{−}) ⊂ {x : λ(x) ⪯ λ}.

Proof. Write

Θ_{λ}^{−} = ∑_{y≤t_{λ}} a_{y}(Q)T̃_{y},

where a_{y}(Q) ∈ ℤ_{+}[Q] (see Lemma 5.1 and Corollary 5.7 of [4]).

Choose a dominant coweight μ′ such that μ′ + λ(x) is also dominant for any x in the support of Θ_{λ}^{−}. Thus we have

T̃_{−(μ′+λ)}^{−1} = Θ_{μ′+λ}^{−} = Θ_{μ′}^{−}Θ_{λ}^{−} = ∑_{y} a_{y}(Q)T̃_{−μ′}^{−1} T̃_{y}.

Let y ∈ supp(Θ_{λ}^{−}). We claim that t_{μ′}y belongs to the support of T̃_{−μ′}^{−1} T̃_{y}. Indeed, under the specialization map H → ℤ[W̃] determined by q^{1/2} ↦ 1, the element T̃_{−μ′}^{−1} T̃_{y} maps to t_{μ′}y. Since no cancellation occurs on the right hand side above, we see from this that t_{μ′}y ∈ supp(T̃_{−(μ′+λ)}^{−1}), and thus

t_{μ′+λ(y)}w_{y} = t_{μ′}y ≤ t_{μ′+λ},

where y = t_{λ(y)}w_{y}. Since μ′ + λ(y) and μ′ + λ are both dominant, it is well-known that this implies μ′ + λ(y) ⪯ μ′ + λ. The lemma follows.

In the case where λ is minuscule, this statement can be considerably sharpened; see Corollary 3.5.

We remark that Lemma 2.1 plays a key role in the proof of Theorem 4.1.

2.2.4 R-polynomials

For any y ∈ W̃, let y = s_{1} ⋯ s_{r}τ (s_{i} ∈ S_{a}, τ ∈ Ω) be a reduced expression for y. Then for any x, we can write

T̃_{y^{−1}}^{−1} = ∑_{x∈W̃} R̃_{x,y}(Q)T̃_{x}.

where R̃_{x,y}(Q) ∈ ℤ[q^{1/2}, q^{−1/2}]. These coefficients R̃_{x,y}(Q) can be thought of as polynomial expressions in Q (as the notation suggests) because of the identity

T̃_{y^{−1}}^{−1} = (T̃_{s_{1}} + Q) ⋯ (T̃_{s_{r}} + Q)T̃_{τ} .

3 The minuscule case

We say λ ∈ X_{∗} is minuscule if ⟨α, λ⟩ ∈ {0, ±1}, for every root α ∈ R. Such coweights are the concern of this section.

The purpose of this section is to present an analogue of Proposition 4.4 from [5] using Θ_{λ}^{−} instead of Θ_{λ}. For simplicity, the theorem is given here for affine Hecke algebras with trivial parameter systems. The generalization to arbitrary parameter systems is straightforward (see [5] for notation and details). Similar arguments to those appearing here apply to Θ_{λ}, giving a short proof of Proposition 4.4 from [5].

Theorem 3.1 Let μ^{−} be minuscule and antidominant, and λ ∈ W_{0}(μ^{−}). Then

Θ_{λ}^{−} = ∑_{x : λ(x)=λ} R̃_{x,t_{λ}}(Q)T̃_{x}.

We begin with some lemmas. For a proof of the first lemma, refer to Proposition 3.4 of [5], where a similar result is given (see also the proof of Corollary 6.6).

Lemma 3.2 Let μ^{−} be an antidominant and minuscule coweight, and let τ ∈ Ω be the unique element such that t_{μ^{−}} ∈ W_{a}τ. Let λ ∈ W_{0}(μ^{−}). Suppose that λ − μ^{−} is a sum of p simple coroots (0 ≤ p ≤ l(t_{μ^{−}}) = r). Then there exists a sequence of simple roots α_{1}, …, α_{p} such that the following hold (setting s_{i} = s_{α_{i}}):

(1) ⟨α_{i}, s_{i−1} ⋯ s_{1}(μ^{−})⟩ = −1, ∀1 ≤ i ≤ p;

(2) there is a reduced expression for t_{μ^{−}} of the form t_{μ^{−}} = s_{1} ⋯ s_{p}t_{1} ⋯ t_{r−p}τ;

(3) there is a reduced expression for t_{λ} of the form t_{λ} = t_{1} ⋯ t_{r−p}(^{τ}s_{1}) ⋯ (^{τ}s_{p})τ;

(4) Θ_{λ}^{−} = T̃_{t_{1}} ⋯ T̃_{t_{r−p}} T̃_{^{τ}s_{1}}^{−1} ⋯ T̃_{^{τ}s_{p}}^{−1}T̃_{τ}, where t_{j} ∈ S_{a}, ∀j ∈ {1, …, r − p}.

Lemma 3.3 Let x ∈ W̃, and suppose that xs_{α} > x for all α ∈ Π. Then l(xw) = l(x) + l(w) for all w ∈ W_{0}.

Since μ^{−} is antidominant, this lemma applies to the expression t_{μ^{−}} = s_{1} ⋯ s_{p}t_{1} ⋯ t_{r−p}τ. It then applies to the expression t_{1} ⋯ t_{r−p}τ as well. It follows that we can think of the formula in Lemma 3.2(4) as

Θ_{λ}^{−} = T̃_{w^{λ}} T̃_{s_{1}}^{−1} ⋯ T̃_{s_{p}}^{−1} = T̃_{w^{λ}} T̃_{w^{−1}}^{−1}

where t_{λ} = w^{λ}w, with w ∈ W_{0} and w^{λ} the minimal length representative for the coset t_{λ}W_{0}. This observation is helpful towards proving the main result of this section.

Lemma 3.4 For λ, s_{1}, …, s_{p}, and t_{1}, …, t_{r−p} as in Lemma 3.2, the mapping

{y : y ≤ s_{1} ⋯ s_{p}} ⟶ {x : λ(x) = λ and x ≤ t_{λ}}

defined by

y ↦ t_{1} ⋯ t_{r−p}τy = w^{λ}y

is bijective.

Proof of Theorem 3.1. Let w = s_{1} ⋯ s_{p}, and w^{λ} = t_{1} ⋯ t_{r−p}τ, so that t_{μ^{−}} = ww^{λ} and t_{λ} = w^{λ}w. We have

T̃_{s_{1}}^{−1} ⋯ T̃_{s_{p}}^{−1} = ∑_{y : y≤w} R̃_{y,w}(Q)T̃_{y}

The expression for Θ_{λ}^{−} of Lemma 3.2, together with the fact that T̃_{w^{λ}} T̃_{y} = T̃_{w^{λ}y} for all y ∈ W_{0} (since l(w^{λ}y) = l(w^{λ}) + l(y)), implies

Θ_{λ}^{−} = ∑_{y : y≤w} R̃_{y,w}(Q)T̃_{w^{λ}y}.

Using the recursion formula of Lemma 2.5 (1) from [5], we obtain R̃_{y,w}(Q) = R̃_{w^{λ}y,t_{λ}}(Q). In view of the bijection given in Lemma 3.4, we have

∑_{y : y≤w} R̃_{w^{λ}y,t_{λ}}(Q)T̃_{w^{λ}y} = ∑_{x : λ(x)=λ} R̃_{x,t_{λ}}(Q)T̃_{x},

which completes the proof.

For the minuscule case, Theorem 3.1 yields the following improvement on Lemma 2.1.

Corollary 3.5 Let λ ∈ X_{∗} be minuscule. Then

supp(Θ_{λ}^{−}) = {x : λ(x) = λ and x ≤ t_{λ}}.

Here we have used Lemma 2.5 (5) of [5], which asserts that R̃_{x,y}(Q) ≠ 0 if and only if x ≤ y. The Bernstein function z_{μ} has a very simple form when μ is minuscule (cf. Theorem 4.3 of [5]):

Corollary 3.6 If μ is dominant and minuscule, then

z_{μ} = ∑_{x∈Adm(μ)} R̃_{x,t_{λ(x)}}(Q)T̃_{x}.

4 Multiples of the Drinfeld case

Fix positive integers n and m, and an integer 1 ≤ k ≤ n. In this section, we establish in Theorem 4.1 a formula for the Θ_{λ}^{−} functions of GL_{n} when λ = me_{k} (where e_{k} denotes the coweight of GL_{n} with kth coordinate equal to 1, and all other coordinates equal to 0).

In this section, we adopt the following notation: for 1 ≤ i ≤ n − 1, let α_{i} = α̌_{i} = e_{i} − e_{i+1}, and let s_{i} = s_{α_{i}}. We single out the element τ ∈ Ω given by τ = t_{(1,0,...,0)}s_{1} ⋯ s_{n−1}.

Theorem 4.1 For the coweight me_{k} of GL_{n}, we have

Θ_{me_{k}}^{−} = ∑_{x : λ(x)⪯me_{k}} R̃_{x,t_{me_{k}}} (Q)T̃_{x}.

Consequently, we have a result for me_{k} analogous to that of Corollary 3.5 for λ minuscule, that is,

supp(Θ_{me_{k}}^{−}) = {x : λ(x) ⪯ me_{k} and x ≤ t_{me_{k}}}.

We also get the following explicit formula for the Bernstein function z_{me_{1}}, analogous to Corollary 3.6: Corollary 4.2

z_{me_{1}} = ∑_{x∈Adm(μ)} ( ∑_{λ(x)⪯λ , x≤t_{λ}} R̃_{x,t_{λ}}(Q)) T̃_{x},

where the inner sum ranges over λ ∈ W_{0}(me_{1}) such that λ(x) ⪯ λ and x ≤ t_{λ}.

We require three lemmas before the proof of the theorem. In the following arguments we use the notation ∏ to denote products even though we are working in a non-commutative ring. We will use the following convention: ∏_{i=1}^{n} a_{i} will denote the product a_{1}a_{2} ⋯ a_{n} (in that order). Lemma 4.3 For the coweight me_{k} of GL_{n}, we have

Θ_{me_{k}}^{−} = (T̃_{s_{k−1}} ⋯ T̃_{s_{1}} T̃_{τ} T̃_{s_{n−1}}^{−1} ⋯ T̃_{s_{k}}^{−1})^{m}.

Proof. From Lemma 3.2, we have

Θ_{e_{k}}^{−} = T̃_{s_{k−1}} ⋯ T̃_{s_{1}} T̃_{τ} T̃_{s_{n−1}}^{−1} ⋯ T̃_{s_{k}}^{−1}.

Then the formula for Θ_{me_{k}}^{−} follows from the fact that Θ_{ν_{1}+ν_{2}}^{−} = Θ_{ν_{1}}^{−}Θ_{ν_{2}}^{−} for all ν_{1}, ν_{2} ∈ X_{∗}.

Lemma 4.4 Let w, y ∈ W̃. Then

T̃_{w}T̃_{y} = ∑_{x=w̃ỹ}a_{x}T̃_{x}

where w̃ and ỹ range over certain subexpressions of w and y, respectively. Proof. This is an easy induction on the length of y.

Lemma 4.5 Let x be a subexpression of

t_{me_{k}} = (s_{k−1} ⋯ s_{1}τs_{n−1} ⋯ s_{k})^{m}

such that for some 1 ≤ i ≤ k − 1, at least one s_{i} is deleted. Then λ(x) ⪯̸ me_{k}. Proof. We can write

x = u_{1}τv_{1} ⋯ u_{m}τv_{m}

for suitable subexpressions u_{1}, …, u_{m} and v_{1}, …, v_{m} of s_{k−1} ⋯ s_{1} and s_{n−1} ⋯ s_{k}, respectively. Suppose that p is the least index such that u_{p} ≠ s_{k−1} ⋯ s_{1}. Then

x = ( ∏_{i=1}^{p−1} t_{e_{k}}s_{k} ⋯ s_{n−1}v_{i})(t_{e_{j}}u_{p}s_{1} ⋯ s_{n−1}v_{p})( ∏_{i=p+1}^{m} u_{i}τv_{i}),

for some j with 1 ≤ j < k. Since s_{k} ⋯ s_{n−1}v(e_{j}) = e_{j} for any subexpression v of s_{n−1} ⋯ s_{k},

( ∏_{i=1}^{p−1} t_{e_{k}}s_{k} ⋯ s_{n−1}v_{i})(t_{e_{j}})( ∏_{i=1}^{p−1} t_{e_{k}}s_{k} ⋯ s_{n−1}v_{i})^{−1} = t_{e_{j}}.

It follows that the translation part λ(x) is the sum of e_{j} and a non-negative integral linear combination of vectors e_{i} (i = 1, …, n). Indeed, the translation part of

( ∏_{i=1}^{p−1} t_{e_{k}}s_{k} ⋯ s_{n−1}v_{i})(u_{p}s_{1} ⋯ s_{n−1}v_{p})( ∏_{i=p+1}^{m} u_{i}τv_{i})

is necessarily a vector (b_{1}, …, b_{n}) where b_{i} ∈ ℤ_{+} for every i.

We thus see that one of the first k − 1 coordinates of λ(x) is positive (namely the j-th coordinate is), and this implies that λ(x) ⪯̸ me_{k}.

Proof of Theorem 4.1. Let E = ({0, 1}^{n−1})^{m} − ({1}^{k−1} × {0, 1}^{n−k})^{m}. Using T̃_{s}^{−1} = T̃_{s} + Q and expanding the left hand side, we can write

T̃_{−me_{k}}^{−1} = Θ_{me_{k}}^{−} + ∑_{ϵ∈E} Q^{m(n−1)−σ(ϵ)} ∏_{i=1}^{m} T̃_{s_{k−1}}^{ϵ_{k−1}^{i}} ⋯ T̃_{s_{1}}^{ϵ_{1}^{i}} T̃_{τ} T̃_{s_{n−1}}^{ϵ_{n−1}^{i}} ⋯ T̃_{s_{k}}^{ϵ_{k}^{i}}.  (1)

Here ϵ_{j}^{i} ∈ {0, 1} and we denote ϵ = (ϵ^{1}, …, ϵ^{m}), and ϵ^{i} = (ϵ_{1}^{i}, …, ϵ_{n−1}^{i}), i = 1, …, m, and σ(ϵ) = ∑_{i=1}^{m} ∑_{j=1}^{n−1} ϵ_{j}^{i}.

From Lemma 2.1, we know that supp(Θ_{me_{k}}^{−}) ⊂ {x : λ(x) ⪯ me_{k}}. Thus we need only prove that if T̃_{x} is in the support of the second term on the right hand side, then λ(x) ⪯̸ me_{k}. Indeed, then the first and second terms on the right hand side of (1) have disjoint supports, and so the coefficients of like terms will be equal in T̃_{−me_{k}}^{−1} and Θ_{me_{k}}^{−}.

Let ϵ = (ϵ^{1}, …, ϵ^{m}) ∈ E, and consider

∏_{i=1}^{m} T̃_{s_{k−1}}^{ϵ_{k−1}^{i}} ⋯ T̃_{s_{1}}^{ϵ_{1}^{i}} T̃_{τ} T̃_{s_{n−1}}^{ϵ_{n−1}^{i}} ⋯ T̃_{s_{k}}^{ϵ_{k}^{i}}.

By Lemma 4.4, if x is in the support of this product, then x is a subexpression of

∏_{i=1}^{m} s_{k−1}^{ϵ_{k−1}^{i}} ⋯ s_{1}^{ϵ_{1}^{i}}τs_{n−1}^{ϵ_{n−1}^{i}} ⋯ s_{k}^{ϵ_{k}^{i}}.

Since E excludes the elements of ({1}^{k−1} ×{0, 1}^{n−k})^{m}, we know that for some 1 ≤ i ≤ m and 1 ≤ j ≤ k −1, that ϵ_{j}^{i} = 0. But this is equivalent to the deletion of some s_{j} (1 ≤ j ≤ k − 1) from the expression t_{me_{k}} = (s_{k−1} ⋯ s_{1}τs_{n−1} ⋯ s_{k})^{m}. By Lemma 4.5, we have λ(x) ⪯̸ me_{k}, and the proof is complete.

5 Minimal expressions

We say Θ_{λ}^{−} has a minimal expression if it can be written in the form

Θ_{λ}^{−} = T̃_{t_{1}}^{ϵ_{1}} ⋯ T̃_{t_{r}}^{ϵ_{r}} T̃_{τ} ,

where t_{λ} = t_{1} ⋯ t_{r}τ (t_{i} ∈ S_{a}, τ ∈ Ω) is a reduced expression and ϵ_{i} ∈ {1, −1} for every 1 ≤ i ≤ r. Such expressions played a key role in Theorems 3.1 and 4.1.

Lemma 3.2 asserts that Θ_{λ}^{−} has a minimal expression whenever λ is minuscule. If λ is any coweight for GL_{n}, then we may write

λ = λ_{1} + ⋯ + λ_{k}

where each λ_{j} is minuscule and

l(t_{λ}) = l(t_{λ_{1}}) + ⋯ + l(t_{λ_{k}}).

It follows that for any coweight λ of GL_{n}, there is a minimal expression for Θ_{λ}^{−}. Letting w^{λ_{i}} denote the minimal representative for the coset t_{λ_{i}}W_{0} and writing t_{λ_{i}} = w^{λ_{i}}w_{i} (w_{i} ∈ W_{0}), we may recover a minimal expression for

Θ_{λ}^{−} = T̃_{w^{λ_{1}}} T̃_{w_{1}^{−1}}^{−1} ⋯ T̃_{w^{λ_{k}}} T̃_{w_{k}^{−1}}^{−1}

by choosing reduced expressions for every w^{λ_{i}} and w_{i}.

Clearly a similar result would follow for any root system with the property that Θ_{ω}^{−} has a minimal expression for every Weyl conjugate ω of every fundamental coweight. It seems to be an interesting combinatorial problem to determine the root systems (besides that for GL_{n}) which satisfy this property.

In principle, a minimal expression for Θ_{λ}^{−} allows one to write it as an explicit linear combination of the Iwahori-Matsumoto generators T_{w}, simply by using the formula T̃_{s}^{−1} = T̃_{s} + Q and expanding the product. The result is a linear combination of certain products

T_{s_{1}} ⋯ T_{s_{g}}T_{σ}

(s_{i} ∈ S_{a}, σ ∈ Ω), where s_{1} ⋯ s_{g}σ ranges over certain subexpressions of t_{λ} (which subexpressions occur is governed by the signs ϵ_{j} in the minimal expression). These may in turn be simplified by using the well-known formula

T_{s_{1}} ⋯ T_{s_{g}} = ∑_{w} N(\underline{s}, w, q)T_{w},

(cf. [11], Lemma 3.7). Here \underline{s} = (s_{1}, …, s_{g}) and N(\underline{s}, w, q) is the number of 𝔽_{q}-rational points on the variety Z(\underline{s}, w) consisting of all sequences (I_{1}, …, I_{g}) where the I_{i} are Iwahori subgroups of G^{sc}(𝔽̄_{q}((t))) (here G^{sc} is the simply connected group associated to the given root system) such that the relative positions of adjacent subgroups satisfy

inv(I_{i−1}, I_{i}) = s_{i}

for all 1 ≤ i ≤ g, and I_{g} = wI_{0}w^{−1}, where I_{0} is a fixed “standard” Iwahori subgroup.

We forgo the cumbersome task of describing more completely the resulting expressions for Θ_{λ}^{−} in terms of the generators T_{w}. The combinatorics are best described in the geometric framework of Demazure resolutions. We explain this in the following section.

Remark. Let λ be a coweight for GL_{n}, and write λ = m_{1}e_{1} + ⋯ + m_{n}e_{n}. One finds a similar expression for Θ_{λ}^{−} starting from

Θ_{λ}^{−} = Θ_{m_{1}e_{1}}^{−} ⋯ Θ_{m_{n}e_{n}}^{−},

and making use of Theorem 4.1.

6 Sheaf-theoretic meaning of minimal expressions

The goal of this section is to describe a sheaf-theoretic interpretation of a minimal expression for Θ_{λ}^{−}: the corresponding perverse sheaf on the affine flag variety is the push-forward of an explicit perverse sheaf on a Demazure resolution of the Schubert variety X(t_{λ}). We proceed to illustrate this statement in more detail.

6.1 Affine flag variety

Let k = 𝔽_{q} denote the finite field with q elements, and let k̄ denote an algebraic closure of k. Let G be the split connected reductive group over k whose root system is (X^{∗}, X_{∗}, R, Ř, Π). Choose a split torus T and a k-rational Borel subgroup B containing T, which give rise to R and Π.

Denote by Fl the affine flag variety for G. This is an ind-scheme over k whose k-points are given by

Fl(k) = G(k((t)))/I_{k},

where I = I_{k} ⊂ G(k[[t]]) is the Iwahori subgroup whose reduction modulo t is B.

Fix a prime ℓ ≠ char(k), and make a fixed choice for \sqrt{q} ∈ ℚ̄_{ℓ} (for Tate twists).

Let D^{b}(Fl) denote the category D_{c}^{b}(Fl, ℚ̄_{ℓ}). By definition D_{c}^{b}(Fl, ℚ̄_{ℓ}) is the inductive 2-limit of categories D_{c}^{b}(X, ℚ̄_{ℓ}) where X ⊂ Fl ranges over all projective k-schemes which are closed subfunctors of the ind-scheme Fl. The category D_{c}^{b}(X, ℚ_{ℓ}) is the “derived” category of Deligne [3]: ℚ_{ℓ}⊗ the projective 2-limit of the categories D_{ctf}^{b}(X, ℤ/ℓ^{n}ℤ). For any finite extension E of ℚ_{ℓ} contained in ℚ̄_{ℓ}, the definition of D_{c}^{b}(X, E) is similar, and by definition D_{c}^{b}(X, ℚ̄_{ℓ}) is the inductive 2-limit of the categories D_{c}^{b}(X, E).

For f : X → Y a morphism of finite-type k-schemes, we have the four “derived” functors f_{∗}, f_{!} : D_{c}^{b}(X, ℚ̄_{ℓ}) → D_{c}^{b}(Y, ℚ̄_{ℓ}) and f^{∗}, f^{!} : D_{c}^{b}(Y, ℚ̄_{ℓ}) → D_{c}^{b}(X, ℚ̄_{ℓ}). This notation should cause no confusion, since we never use the non-derived versions of the pull-back and push-forward functors in this paper.

We define the category P_{I}(Fl): it is the full subcategory of D^{b}(Fl) whose objects are I-equivariant perverse sheaves for the middle perversity (by definition the latter have finite dimensional support).

The I-orbits on Fl correspond to W̃. Given w ∈ W̃, we denote by Y (w) = IwI/I the corresponding Bruhat cell, and we denote its closure by X(w) = \overline{Y (w)}. Further, let ℚ̄_{ℓ,w} denote the constant sheaf on Y (w), and define A_{w} = ℚ̄_{ℓ,w}[l(w)](l(w)/2). This is a self-dual perverse sheaf on Y (w).

Let j_{w} : Y (w) ↪ X(w) denote the open immersion. We define J_{w∗} = j_{w∗}A_{w} and J_{w!} = j_{w!}A_{w}. These are perverse sheaves in P_{I}(Fl) satisfying D(J_{w∗}) = J_{w!}. (Here D denotes Verdier duality.)

Given G ∈ P_{I}(Fl) we may define the corresponding function [G] on Fl(k), which we may identify with an element in H:

[G](x) = Tr(Fr_{q}, G_{x}),

where Fr_{q} denotes the Frobenius morphism on Flk̄ (raising coordinates to power q). We have

[J_{w!}] = ε_{w}q_{w}^{−1/2}T_{w}
[J_{w∗}] = ε_{w}q_{w}^{1/2}T_{w^{−1}}^{−1}.

6.2 Convolution of sheaves

Following Lusztig [15], one can define a convolution product ⋆ : P_{I}(Fl) × P_{I}(Fl) → D^{b}(Fl). We formulate this in a way similar to [6]. Given G_{i} ∈ P_{I}(Fl), i = 1, 2, we can choose X(w_{i}) such that the support of G_{i} is contained in X(w_{i}), for i = 1, 2. We may identify Fl with the space of all “affine flags” L for G(k((t))); there is a base point L_{0} whose stabilizer in G(k((t))) is the “standard” Iwahori subgroup I. Then X(w) is identified with the space of all affine flags L such that the relative position between the base point L_{0} and L satisfies

inv(L_{0}, L) ≤ w

in the Bruhat order on W̃. The “twisted” product X(w_{1}) ˜×X(w_{2}) is the space of pairs (L, L′) ∈ Fl × Fl such that

inv(L_{0}, L) ≤ w_{1}
inv(L, L′) ≤ w_{2}.

We can find a finite-dimensional projective subvariety X ⊂ Fl with the property that (L, L′) ∈ X(w_{1}) ˜×X(w_{2}) ⇒ L′ ∈ X. The “multiplication” map m : X(w_{1}) ˜×X(w_{2}) → X given by (L, L′) ↦ L′ is proper.

Now G_{i} (i = 1, 2) determine a well-defined perverse sheaf G_{1} ˜⊠G_{2} on X(w_{1}) ˜×X(w_{2}) (see e.g. [6]). We define

G_{1} ⋆ G_{2} = m_{∗}(G_{1} ˜⊠G_{2}).

The convolution G_{1} ⋆ G_{2} ∈ D^{b}(Fl) is independent of the choice of X(w_{i}) and X.

The object G_{1} ⋆ G_{2} is I-equivariant in a suitable sense, so that we can regard its function Tr(Fr_{q}, G_{1} ⋆ G_{2}) as an element of the Hecke algebra H.

It is well-known that this product is compatible with the function-sheaf dictionary:

[G_{1} ⋆ G_{2}] = [G_{1}] ⋆ [G_{2}].

Here ⋆ on the right hand side is just the usual product in H.

Later we shall use the following fact, referred to in the sequel simply as associativity: if G_{i} (i = 1, 2, 3) are objects of P_{I}(Fl) such that G_{1} ⋆G_{2} ∈ P_{I}(Fl) and G_{2} ⋆G_{3} ∈ P_{I}(Fl), then there is a canonical isomorphism G_{1} ⋆ (G_{2} ⋆ G_{3}) ˜→(G_{1} ⋆ G_{2}) ⋆ G_{3} (the “associativity constraint”). This is proved by identifying each canonically with the “triple product” G_{1} ⋆ G_{2} ⋆ G_{3}, whose construction is similar (see section 6.3).

6.3 Demazure resolution vs. twisted product

It is clear that we can define in a similar way the k-fold convolution product ⋆ : P_{I}(Fl)^{k} → D^{b}(Fl). To do this we define the k-fold twisted product X(w_{1}) ˜× ⋯ ˜×X(w_{k}) to be the space of k-tuples (L_{1}, …, L_{k}) such that

inv(L_{i−1}, L_{i}) ≤ w_{i}

for 1 ≤ i ≤ k. If w = w_{1} ⋯ w_{k} and l(w) = ∑_{i} l(w_{i}), then the multiplication map

m : X(w_{1}) ˜× ⋯ ˜×X(w_{k}) ⟶ X(w)

given by (L_{1}, …, L_{k}) ↦ L_{k} induces an isomorphism on open subschemes

m : Y (w_{1}) ˜× ⋯ ˜×Y (w_{k}) ˜ ⟶Y (w).

If moreover each w_{i} is a simple reflection s_{i}, then the twisted product is smooth, since it is a succession of ℙ^{1}-bundles. We have proved the following lemma.

Lemma 6.1 If w = s_{1} ⋯ s_{r} is a reduced expression, then the twisted product

m : X(s_{1}) ˜× ⋯ ˜×X(s_{r}) ⟶ X(w)

is a Demazure resolution for the Schubert variety X(w).

We use the k-fold twisted product to define the k-fold convolution product: as before, the objects G_{i} ∈ P_{I}(Fl) determine a (unique, perverse) twisted exterior product G_{1} ˜⊠ ⋯ ˜⊠G_{k}, and we set

G_{1} ⋆ ⋯ ⋆ G_{k} = m_{∗}(G_{1} ˜⊠ ⋯ ˜⊠G_{k}).

We have the following generalized associativity constraint. Consider a product with k terms

(⋯ (G_{1} ⋆ (⋯ ⋆ G_{i})) ⋯) ⋆ (⋯ (G_{j} ⋆ (⋯ ⋆ G_{k})) ⋯)

where the placement of the parentheses is arbitrary with the proviso that the product is defined (i.e., at every stage we convolve objects of P_{I}(Fl)). Then this can be identified canonically with the k-fold product

G_{1} ⋆ ⋯ ⋆ G_{k}.

This can be seen easily by induction on k.

6.4 Properties of certain convolutions

The convolution of two I-equivariant perverse sheaves on Fl is not perverse in general. However, the following result of I. Mirkovic (unpublished) shows this conclusion does hold in some important cases. We are grateful to R. Bezrukavnikov, who communicated this result to the first author, and to I. Mirkovic, for his kind permission to include the result in this paper.

In the notation of [1], we let ^{p}D^{≤0}(Fl) (resp. ^{p}D^{≥0}(Fl)) denote the objects P ∈ D^{b}(Fl) whose perverse cohomology sheaves vanish in degree ≥ 1 (resp. ≤ −1). Thus the perverse sheaves on Fl are precisely the objects in ^{p}D^{≥0}(Fl) ∩ ^{p}D^{≤0}(Fl).

Proposition 6.2 (Mirkovic) (a) Let P ∈ P_{I}(Fl). Then for any w ∈ W̃, we have

(1) J_{w!} ⋆ P and P ⋆ J_{w!} belong to ^{p}D^{≥0}(Fl),

(2) J_{w∗} ⋆ P and P ⋆ J_{w∗} belong to ^{p}D^{≤0}(Fl).

(b) In particular, J_{w_{1}∗} ⋆ J_{w_{2}!} and J_{w_{1}!} ⋆ J_{w_{2}∗} are perverse, for every w_{1}, w_{2} ∈ W̃.

Proof. (a),(1). We consider J_{w!} ⋆ P. Suppose P is supported on X(w′) and recall that the convolution is given by

J_{w!} ⋆ P = m_{!}(J_{w!} ˜⊠P),

where m : X(w) ˜×X(w′) → X is as in section 6.2 (note that m_{!} = m_{∗}, since m is proper). We have

J_{w!} ⋆ P = m_{!}(j_{w!}(A_{w}) ˜⊠P)
= [m ∘ (j_{w} ˜×id)]_{!}(A_{w} ˜⊠P).

Note that m ∘ (j_{w} ˜×id) : Y (w) ˜×X(w′) → X is affine, and that A_{w} ˜⊠P is perverse on its source. Therefore J_{w!}⋆P ∈ ^{p}D^{≥0}(Fl) follows from the following general fact (cf. [1], Thm. 4.1.1 and Cor. 4.1.2): If F : X → Y is an affine morphism, then F_{∗} : D_{c}^{b}(X) → D_{c}^{b}(Y) (resp. F_{!}) preserves ^{p}D^{≤0} (resp. ^{p}D^{≥0}).

A similar argument gives P ⋆ J_{w!} ∈ ^{p}D^{≥0}(Fl), once it is noted that m ∘ (id ˜×j_{w}) : X(w′) ˜×Y (w) → X is also an affine morphism.

Part (a),(2) is similar, and part (b) is an immediate consequence of part (a).

Now we let P_{I}(Fl) ∩ J_{w∗}P_{I}(Fl) (resp. P_{I}(Fl) ∩ J_{w!}P_{I}(Fl)) denote the full subcategory of P_{I}(Fl) whose objects are of the form J_{w∗} ⋆ P (resp. J_{w!} ⋆ P) for some P ∈ P_{I}(Fl).

Corollary 6.3 For any w ∈ W̃, we have

J_{w!} ⋆ J_{w^{−1}∗} = J_{w^{−1}∗} ⋆ J_{w!} = J_{e},

where e ∈ W̃ is the identity element. Together with the associativity constraint, these identities imply that

J_{w!} ⋆ − : P_{I}(Fl) ∩ J_{w^{−1}∗}P_{I}(Fl) ⟶ P_{I}(Fl) ∩ J_{w!}P_{I}(Fl)

is an equivalence of categories, with inverse J_{w^{−1}∗} ⋆ −.

Proof. We prove that J_{w!} ⋆ J_{w^{−1}∗} = J_{e} (the other equality is similar). Let X(y) be an irreducible component in the support of P := J_{w!} ⋆ J_{w^{−1}∗}. Since P is perverse and I-equivariant, the restriction P |Y (y) is an I-equivariant ℓ-adic local system on the affine space Y (y). We need to show that y = e and P |Y (e) = ℚ̄_{ℓ}. For the former it is sufficient to prove that y ≠ e implies P |Y (y) = 0.

Since P |Y (y) is an I-equivariant ℓ-adic local system on the I-orbit Y (y), and the stabilizer in I of any point in this orbit is geometrically connected, it follows that P |Y (y) is a constant local system (placed in degree −l(y) when regarded as a complex). Write α_{1}, α_{2}, …, α_{r} for the eigenvalues of Fr_{q} on P |Y (y)[−l(y)], counted with multiplicity. We have the following identity for every n ≥ 1:

J_{e}(y) = Tr(Fr_{q^{n}}, J_{w!}) ⋆ Tr(Fr_{q^{n}}, J_{w^{−1}∗})(y) = Tr(Fr_{q^{n}}, P |Y (y)) = ϵ_{y} ∑_{i=1}^{r} α_{i}^{n}.

We thus have, for every n ≥ 1:

ϵ_{y} ∑_{i=1}^{r} α_{i}^{n} = \begin{cases}0, if y ≠ e \\ 1, if y = e. \end{cases}

The linear independence of characters (or rather its proof) implies that distinct numbers β ∈ ℚ̄_{ℓ}^{×} determine linearly independent characters n ↦ β^{n} on the semi-group of positive integers. Together with the above formula this is enough information to determine the eigenvalues of Fr_{q} on P |(Y (y) (if y ≠ e then r = 0, and if y = e then r = 1 and α_{1} = 1). Thus P |Y (y) = 0 if y ≠ e and P |Y (e) = ℚ̄_{ℓ}, as desired.

It is straightforward to check the following properties. From now on we omit the convolution sign ⋆ in the product of perverse sheaves.

Lemma 6.4 1. If l(xy) = l(x) + l(y), then J_{x!}J_{y!} = J_{xy!}.

2. Under the same assumption, J_{x∗}J_{y∗} = J_{xy∗}.

Proof. The first can be checked from the definitions, and the second follows on applying Verdier duality. We have used that Verdier duality is compatible with convolution: D(G_{1} ⋆ G_{2}) = DG_{1} ⋆ DG_{2}, if G_{i} ∈ P_{I}(Fl) (i = 1, 2).

Remark. We remark that Corollary 6.3 and Lemma 6.4 allow us to perform algebraic manipulations involving the perverse sheaf J_{w!} (resp. J_{w∗}): essentially it behaves just like its function ε_{w}T̃_{w} (resp. ε_{w}T̃_{w^{−1}}^{−1}) (but when multiplying sheaves, one has to take care that they are each perverse). For example, we have the following cancellation property. Let P_{i} ∈ P_{I}(Fl) (i = 1, 2) be such that J_{w∗} ⋆ P_{i} ∈ P_{I}(Fl) (i = 1, 2); then J_{w∗}P_{1} ≅ J_{w∗}P_{2} implies P_{1} ≅ P_{2} (multiply both sides by J_{w^{−1}!} and use associativity). We shall use this several times in the proof of Lemma 6.5 below.

6.5 Sheaf analogue Ξ_{λ}^{−} of Θ_{λ}^{−}

We now define the sheaf-analogue of Θ_{λ}^{−}. We write J_{λ∗} (resp. J_{λ!}) in place of J_{t_{λ}∗} (resp. J_{t_{λ}!}). If λ = λ_{1}−λ_{2}, where λ_{i} is anti-dominant (i = 1, 2), then we define

Ξ_{λ}^{−} = J_{λ_{1}!}J_{−λ_{2}∗}.

By Proposition 6.2, Ξ_{λ}^{−} is an object of P_{I}(Fl). Moreover it clearly satisfies

[Ξ_{λ}^{−}] = ε_{λ}Θ_{λ}^{−}.

By [4], we know that supp(Θ_{λ}^{−}) ⊂ {x | x ≤ t_{λ}}, and hence that Ξ_{λ}^{−} is supported on X(t_{λ}).

Our next goal is to prove the sheaf-theoretic analogues of the relations in the Bernstein presentation of H, in the following lemma. The proof follows Lemma 4.4 of [13] very closely, taking into account Proposition 6.2, Corollary 6.3 and Lemma 6.4. Since a little extra care must be taken in the present context of perverse sheaves, we give detailed arguments for the convenience of the reader.

Lemma 6.5 (1) If x, y ∈ W̃ commute and l(xy) = l(x) + l(y), then J_{x!}J_{y^{−1}∗} = J_{y^{−1}∗}J_{x!}. In particular, if μ, λ ∈ X_{∗} are both dominant or antidominant, then J_{μ!}J_{−λ∗} = J_{−λ∗}J_{μ!}.

(2) Ξ_{λ}^{−} is independent of the choice of λ_{i} ( i = 1, 2 ).

(3) Ξ_{λ}^{−}Ξ_{μ}^{−} = Ξ_{λ+μ}^{−} for λ, μ ∈ X_{∗}.

(4) If s = s_{α} (α ∈ Π) and ⟨α, λ⟩ = 0, then

J_{s∗}Ξ_{λ}^{−} = Ξ_{λ}^{−}J_{s∗}.

Moreover, this object belongs to P_{I}(Fl).

(5) If ⟨α, λ⟩ = −1, then

J_{s∗}Ξ_{λ}^{−}J_{s∗} = Ξ_{sλ}^{−}.

Proof. (1). By Lemma 6.4, we have J_{x!}J_{y!} = J_{xy!} = J_{yx!} = J_{y!}J_{x!}. The result follows by two applications of Corollary 6.3: multiply first on the left and then on the right by J_{y^{−1}∗}. (2). Let λ = λ_{1} − λ_{2} = λ′_{1} − λ′_{2}, where λ_{i}, λ′_{i} are antidominant (i = 1, 2). Then λ′_{2} + λ_{1} = λ′_{1} + λ_{2}, and so J_{λ′_{2}!}J_{λ_{1}!} = J_{λ′_{1}!}J_{λ_{2}!}. Arguing as in (1), this yields J_{λ_{1}!}J_{−λ_{2}∗} = J_{−λ′_{2}∗}J_{λ′_{1}!}. Then (1) yields J_{λ_{1}!}J_{−λ_{2}∗} = J_{λ′_{1}!}J_{−λ′_{2}∗}.

(3). Write λ = λ_{1} − λ_{2} and μ = μ_{1} − μ_{2}, where λ_{i}, μ_{i} are antidominant (i = 1, 2). Then by associativity we have

Ξ_{λ}^{−}Ξ_{μ}^{−} = (J_{λ_{1}!}J_{−λ_{2}∗})(J_{μ_{1}!}J_{−μ_{2}∗})
= J_{λ_{1}!}(J_{−λ_{2}∗}J_{μ_{1}!})J_{−μ_{2}∗}
= J_{λ_{1}!}(J_{μ_{1}!}J_{−λ_{2}∗})J_{−μ_{2}∗}
= (J_{λ_{1}!}J_{μ_{1}!})(J_{−λ_{2}∗}J_{−μ_{2}∗})
= J_{(λ_{1}+μ_{1})!}J_{−(λ_{2}+μ_{2})∗}
= Ξ_{λ+μ}^{−}.

(4). We may write λ = λ_{1} − λ_{2}, where λ_{i} is antidominant and ⟨α, λ_{i}⟩ = 0, for i = 1, 2. Since s commutes with t_{λ_{i}} and l(st_{λ_{i}}) = l(t_{λ_{i}}s) = l(t_{λ_{i}}) + 1, the result follows from (1), Lemma 6.4, and associativity.

We note that J_{s∗}(J_{λ_{1}!}J_{−λ_{2}∗}) = J_{st_{−λ_{2}} ∗}J_{t_{λ_{1}} !} is I-equivariant and perverse, by Proposition 6.2. (5). We may write λ = λ_{1} − λ_{2}, where λ_{i} is antidominant (i = 1, 2), ⟨α, λ_{1}⟩ = −1, and ⟨α, λ_{2}⟩ = 0.

As in the proof of (4) above, we note that J_{s∗}(J_{λ_{1}!}J_{−λ_{2}∗}) and (J_{λ_{1}!}J_{−λ_{2}∗})J_{s∗} are each perverse, so by associativity we may unambiguously write

J_{s∗}Ξ_{λ}^{−}J_{s∗} = J_{s∗}(J_{λ_{1}!}J_{−λ_{2}∗})J_{s∗}.

Using (4) and associativity, this is (J_{s∗}J_{λ_{1}!}J_{s∗})J_{−λ_{2}∗} = (J_{s∗}Ξ_{λ_{1}}^{−}J_{s∗})Ξ_{−λ_{2}}^{−}. (Note J_{s∗}J_{λ_{1}!}J_{s_{∗}} is unambiguous and perverse, since st_{λ_{1}} < t_{λ_{1}} implies that J_{λ_{1}!}J_{s∗} = J_{s!}J_{st_{λ_{1}} !}J_{s∗}, and therefore J_{s∗}J_{λ_{1}!}J_{s∗} = J_{st_{λ_{1}} !}J_{s∗}.)

Since by (3) Ξ_{sλ_{1}}^{−}Ξ_{−λ_{2}}^{−} = Ξ_{sλ}^{−}, we are reduced to proving J_{s∗}Ξ_{λ_{1}}^{−}J_{s∗} = Ξ_{sλ_{1}}^{−}, i.e., to prove the result for general λ it is enough to consider λ which are antidominant.

Therefore assume λ is antidominant, and write l(t_{λ}) = l. Following Lemma 4.4 (b) of [13] we see

• l(t_{λ}s) = l + 1 and l(st_{λ}) = l − 1,

• λ + sλ is antidominant,

• l(t_{λ}st_{λ}) = 2l − 1 and l(t_{λ}st_{λ}s) = 2l − 2; in particular l(t_{λ}st_{λ}) = l(t_{λ}) + l(st_{λ}).

Taking these relations, the previous parts of the Lemma, Corollary 6.3, and associativity into account, we find

J_{λ!}Ξ_{sλ}^{−} = Ξ_{λ}^{−}Ξ_{sλ}^{−}
= Ξ_{λ+sλ}^{−}
= J_{t_{λ}st_{λ}s!}
= J_{t_{λ}st_{λ}!}J_{s∗}
= J_{λ!}J_{st_{λ}!}J_{s∗}
= J_{λ!}(J_{s∗}J_{λ!}J_{s∗}).

Using Corollary 6.3 again to cancel J_{λ!} from each side, we obtain the desired equality J_{s∗}Ξ_{λ}^{−}J_{s∗} = Ξ_{sλ}^{−}.

Note that property (5) is the analogue of Bernstein’s relation

T̃_{s}^{−1}Θ_{λ}^{−}T̃_{s}^{−1} = Θ_{sλ}^{−},

which was a main ingredient in the proof of Lemma 3.2. In fact the same argument can be applied to prove the following corollary.

Corollary 6.6 If λ is minuscule and t_{λ} = w^{λ}w as in section 3, then

Ξ_{λ}^{−} = J_{w^{λ}!}J_{w∗}.

Writing w^{λ} = t_{1} ⋯ t_{r−p}τ and w = s_{1} ⋯ s_{p} as in section 3, we have

Ξ_{λ}^{−} = J_{t_{1}!} ⋯ J_{t_{r−p}!}J_{τ!}J_{s_{1}∗} ⋯ J_{s_{p}∗}.

Proof. Suppose λ is in the Weyl orbit of an antidominant minuscule coweight μ^{−}. We have

Ξ_{μ^{−}}^{−} = J_{μ^{−}!}.

Choose the sequence of simple reflections s_{1}, …, s_{p} as in Lemma 3.2. By induction on p, we easily see that

Ξ_{s_{p}···s_{1}(μ^{−})}^{−} = J_{t_{1}···t_{r−p}τ!}J_{s_{1}···s_{p}∗}.

Indeed, using induction and Lemma 6.4 this equality for p − 1 can be written

Ξ_{s_{p−1}···s_{1}(μ^{−})}^{−} = J_{s_{p}!}J_{t_{1}···t_{r−p}τ!}J_{s_{1}···s_{p−1}∗}.

Multiplying on each side by J_{s_{p}∗} and using Lemma 6.5 (5), Corollary 6.3, and associativity yields

Ξ_{s_{p}···s_{1}(μ^{−})}^{−} = J_{t_{1}···t_{r−p}τ!}J_{s_{1}···s_{p}∗},

as desired.

The second statement follows from the first, using Lemma 6.4. To justify this, we need to show that

(J_{t_{1}!} ⋯ J_{t_{r−p}!}J_{τ!})(J_{s_{1}∗} ⋯ J_{s_{p}∗}) = J_{t_{1}!} ⋯ J_{t_{r−p}!}J_{τ!}J_{s_{1}∗} ⋯ J_{s_{p}∗},

where the products of the form J_{t_{1}!} ⋯ J_{s_{p}∗} denote the k-fold convolution mentioned in section 6.3. This results from the generalized associativity discussed there.

We remark that it is important here that the underlying expression t_{1} ⋯ t_{r−p}τs_{1} ⋯ s_{p} is reduced.

Taking Corollary 6.6 as well as Lemmas 6.1, 6.4 and 6.5 into account, we get the following sheaf-theoretic interpretation for minimal expressions for Θ_{λ}^{−}.

Theorem 6.7 (a) Let λ be a minuscule coweight of any root system. Write t_{λ} = w^{λ}w = t_{1} ⋯ t_{r−p}τs_{1} ⋯ s_{p}, as in Lemma 3.2. Then

Ξ_{λ}^{−} = m_{∗}(D),

where D is the perverse sheaf

D = (j_{t_{1}!}A_{t_{1}}) ˜⊠ ⋯ ˜⊠(j_{t_{r−p}!}A_{t_{r−p}}) ˜⊠(j_{τ!}A_{τ}) ˜⊠(j_{s_{1}∗}A_{s_{1}}) ˜⊠ ⋯ ˜⊠(j_{s_{p}∗}A_{s_{p}})

on the Demazure resolution m : X(t_{1}) ˜× ⋯ ˜×X(t_{r−p}) ˜×X(τ) ˜×X(s_{1}) ˜× ⋯ ˜×X(s_{p}) → X(t_{λ}) of the Schubert variety X(t_{λ}).

(b) Let λ be a coweight for GL_{n}, and write it as

λ = λ_{1} + ⋯ + λ_{k},

where each λ_{i} is minuscule, and l(t_{λ}) = ∑_{i} l(t_{λ_{i}}). For each i = 1, …, k we have a decomposition and reduced expression

t_{λ_{i}} = w^{λ_{i}}w_{i} = t_{1}^{i} ⋯ t_{q_{i}}^{i}τ^{i}s_{1}^{i} ⋯ s_{p_{i}}^{i},

as in Lemma 3.2.

Then Ξ_{λ}^{−} = m_{∗}(D), where D is the perverse sheaf

J_{t_{1}^{1}!} ˜⊠ ⋯ ˜⊠J_{t_{q_{1}}^{1} !} ˜⊠J_{τ^{1}!} ˜⊠J_{s_{1}^{1}∗} ˜⊠ ⋯ ˜⊠J_{s_{p_{1}}^{1} ∗} ˜⊠ · · · · · · · ˜⊠J_{t_{1}^{k}!} ˜⊠ ⋯ ˜⊠J_{t_{q_{k}}^{k}!} ˜⊠J_{τ^{k}!} ˜⊠J_{s_{1}^{k}∗} ˜⊠ ⋯ ˜⊠J_{s_{p_{k}}^{k} ∗}

on the Demazure resolution m : X̃(t_{λ}) → X(t_{λ}) corresponding to the reduced expression

t_{λ} = (t_{1}^{1} ⋯ t_{q_{1}}^{1}τ^{1}s_{1}^{1} ⋯ s_{p_{1}}^{1}) ⋯ (t_{1}^{k} ⋯ t_{q_{k}}^{k}τ^{k}s_{1}^{k} ⋯ s_{p_{k}}^{k}).

Consequently, Ξ_{λ}^{−} is the push-forward of an explicit perverse sheaf on a Demazure resolution of X(t_{λ}), for every coweight λ of GL_{n}.

Proof. (a). This follows directly from Corollary 6.6 and the definition of the r-fold convolution product. (b). By Lemma 6.5 (3) and generalized associativity, we have

Ξ_{λ}^{−} = Ξ_{λ_{1}}^{−} ⋆ ⋯ ⋆ Ξ_{λ_{k}}^{−},

where the right hand side denotes the k-fold convolution product.

Part (b) then follows from part (a) and another application of generalized associativity.

For x ≤ t_{λ}, write Θ_{λ}^{−}(x) for the coefficient of T_{x}.

Corollary 6.8 Let λ be a minuscule coweight of a root system, or an arbitrary coweight for GL_{n}. Let m, D be as in Theorem 6.7. Then

Θ_{λ}^{−}(x) = ε_{λ}Tr(Fr_{q}, H^{•}(m^{−1}(x), D)),

where the right hand side is the alternating trace of Frobenius on the étale cohomology of the fiber over x with coefficients in D.

7 Acknowledgements

We thank I. Mirkovic for kindly allowing us to include his unpublished result (Proposition 6.2), which was crucial to the last section of this paper. We are also grateful to R. Bezrukavnikov for some very helpful discussions concerning Mirkovic’s theorem.

Some of this paper was written during the first author’s 2000-2001 visit to the Institute for Advanced Study in Princeton, which he thanks for support and hospitality. His research is partially supported by an NSERC research grant and by NSF grant DMS 97-29992.

The second author was supported by an NSERC Undergraduate Student Research Award during the summers of 2000 and 2001.

We wish to thank M. Rapoport for his comments on this paper. We are also grateful to R. Kottwitz for his careful reading and for pointing out some errors in the first version of this paper. We thank him for several other mathematical and expositional suggestions, in particular concerning the proof of Corollary 6.3.

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