Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds

Ulrich Görtz and Thomas J. Haines ^{∗}

Abstract

We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson-Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an earlier paper, which encode the data of the Jordan-Hoelder series. We prove the multiplicity functions are polynomials in q, and we give a sharp bound for their degrees. Our results apply as well to the nearby cycles in the p-adic deformation of Laumon-Haines-Ngô, and also to Wakimoto sheaves.

1 Introduction

Let G be a split connected reductive group over a finite field 𝔽_{p} with algebraic closure k, fix an Iwahori subgroup B ⊂ G(𝔽_{p}([[t]])) and let Fl = G(k((t)))/B_{k} denote the affine flag variety of G. Let q denote a power of p and fix a prime ℓ ≠ p. In [GH], the authors study the Jordan-Hölder series for objects in the Hecke category P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}). This is the category of B-equivariant perverse Weil-sheaves F on Fl having the property that for any x ∈ Fl(𝔽_{q}), the trace of the Frobenius Fr_{q} on the stalk F_{x} satisfies

Tr(Fr_{q}, F_{x}) ∈ ℤ[q, q^{−1}].

Put more precisely, for each x there exists a polynomial P (X, Y) ∈ ℤ[X, Y] such that Tr(Fr_{q^{n}}, F_{x}) = P (q^{n}, q^{−n}), for all positive integers n.

By [GH], §4 the objects in P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) are mixed and have finite length, and the irreducible objects are the integral Tate-twists IC_{w}(i) of the intersection complexes IC_{w} = j_{w,!∗}(\overline{ℚ}_{ℓ})[ℓ(w)]. Here for w belonging to the extended affine Weyl group W̃ we take the Goresky-MacPherson middle extension along the immersion j_{w} : B_{w} ↪ B of the corresponding Schubert cell B_{w}. We may thus define non-negative multiplicities m(F, w, i) by the identity in P_{q}^{B}(Fl, \overline{ℚ}_{ℓ})

F^{ss} = ⨁_{w∈W̃} ⨁_{i∈ℤ}IC_{w}(−i)^{⊕m(F,w,i)}.

∗Research of the second author partially supported by NSF grant DMS 0303605 and a Sloan Research Fellowship.

We formally encode the data of the integers m(F, w, i) with the multiplicity function

m(F, w) := ∑_{i} m(F, w, i)q^{i} ∈ ℤ[q, q^{−1}].

In [GH] we studied the functions m(RΨ, w) for certain nearby cycles sheaves RΨ on Fl which arise as follows. If G is the group GL_{n} or GSp_{2n}, there is a ℤ_{p}-ind-scheme M which is a deformation of the affine Grassmannian Grass_{ℚ_{p}} to the affine flag variety Fl_{𝔽_{p}} for the underlying group G. The ind-scheme M has been constructed over ℤ_{p} for these two groups in [HN1], and it is related to the bad reduction of certain Shimura varieties (see below). In the function field case the analogous deformation FL_{X} over a curve X exists for all split connected reductive groups, and has been studied by Gaitsgory [Ga]. In that case, there is a distinguished point x_{0} ∈ X, and the fiber of FL_{X} over x ∈ X is Fl_{k} if x = x_{0} and is Grass_{k} × G/B if x ≠ x_{0}, where G/B is the flag variety which is the “reduction modulo t” of G(k[[t]])/B_{k}. Let Q_{μ} ⊂ Grass denote the stratum indexed by a dominant coweight μ of G, and let IC_{μ} denote the intersection complex j_{μ,!∗}(\overline{ℚ}_{ℓ})[dim Q_{μ}]. Let RΨ_{μ} = RΨ(IC_{μ}), the nearby cycles being taken with respect to the deformation M or FL_{X}. More precisely, in the p-adic setting we write RΨ_{μ} = RΨ^{M}(IC_{μ}), and in the function-field setting we write RΨ_{μ} = RΨ^{FL^{X}} (IC_{μ} ⊠ δ), where δ is the skyscraper sheaf supported at the base point of G/B.

The sheaf RΨ_{μ} belongs to P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) (see [GH],§2, 6), a fact that is essentially a consequence of the Kottwitz conjecture, see (4.1). Hence we may consider the multiplicity functions m(RΨ_{μ}, w) for w ∈ W̃. By [GH], §7 it is known that m(RΨ_{μ}, w) ≠ 0 if and only if w ∈ Adm(μ), the finite subset of μ-admissible elements in W̃ (see [KR],[HN2] for explicit descriptions of Adm(μ)).

Our main theorem is the following result conjectured in [GH] and proved there for the case G = GL_{n} or μ minuscule.

Theorem 1.1 For any w ∈ Adm(μ), the function m(RΨ_{μ}, w) is a polynomial in q having degree at most ℓ(μ) − ℓ(w). Equivalently, the same statement holds for the functions Tr(Fr_{q}, (RΨ_{μ})_{w}). Equivalently, for each closed point z ∈ B_{w}, the weights of the complex RΨ_{μ}(\frac{ℓ(μ)}{2})_{z} are ≤ ℓ(μ) − ℓ(w).

Here the length function ℓ : W̃ → ℤ_{≥0} is that defined by the simple affine reflections S_{aff} through the walls of the alcove in the Bruhat-Tits building which is fixed by the Iwahori B (see [GH]). The symbol ℓ(μ) denotes the length of the translation element t_{μ} ∈ W̃ corresponding to the cocharacter μ.

The study of m(RΨ, w) relates to nearby cycles on certain Shimura varieties with Iwahori level structure at p. In fact, for such a Shimura variety Sh = Sh(G, h)_{K} where the p-adic group G_{ℚ_{p}} is either GL_{n}×𝔾_{m} or GSp_{2n}, the Rapoport-Zink local model M^{loc} for the singularities in the special fiber Sh_{\overline{𝔽}_{p}} can be realized as the scheme-theoretic closure in M of Q_{μ} ⊂ Grass_{ℚ_{p}}, where μ is the minuscule coweight coming from the Shimura datum h. See [GH], §8 for a more detailed discussion of the connection with Shimura varieties.

A related result concerns the Wakimoto sheaves M_{u,v} on Fl. Here for elements u, v ∈ W̃ we define following I. Mirković the sheaf

M_{u,v} = j_{u!}(\overline{ℚ}_{ℓ})[ℓ(u)] ⋆ j_{v∗}(\overline{ℚ}_{ℓ})[ℓ(v)],

using the standard convolution operation ⋆ : P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) × P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) → D_{c}^{b,Weil}(Fl, \overline{ℚ}_{ℓ}), the latter being the category consisting of objects of the “derived category of ℓ-adic sheaves” D_{c}^{b}(Fl, \overline{ℚ}_{ℓ}), equipped with a Weil structure. A result of Mirković states that M_{u,v} is perverse, hence it belongs to P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) (Mirković’s proof appears in [HP] and in [AB]). Let us renormalize and define

M̃_{u,v} = M_{u,v}(\frac{ℓ(u) + ℓ(v) − ℓ(uv)}{2}).

It is known that the function Tr(Fr_{q}, M_{u,v}) is supported on a subset of the elements {x ∈ W̃ | x ≤ uv}, where the Bruhat order ≤ is defined using S_{aff} (see [H01]).

Theorem 1.2 For any w ≤ uv, the function m(M̃_{u,v}, w) is a polynomial in q having degree at most ℓ(uv)−ℓ(w). Equivalently, the same holds for the functions Tr(Fr_{q}, (M̃_{u,v})_{w}). Equivalently, for each closed point z ∈ B_{w}, the weights of the complex M̃_{u,v}(\frac{ℓ(uv)}{2})_{z} are ≤ ℓ(uv)−ℓ(w).

As we explain at the end of section 4, Theorem 1.2 can be used to prove Theorem 1.1. But in section 3 we will give a more direct proof of Theorem 1.1 using an approach we introduced (but did not complete) in [GH], §10.

Theorem 1.2 might be interesting in its own right, since Wakimoto sheaves have played an important role in several recent papers which study the geometry relating the two presentations (Iwahori-Matsumoto and Bernstein) of an affine Hecke algebra. See [HP], and especially [AB] and [ABG].

2 From bounds on weights to bounds on degrees

Let supp(F) = {x ∈ W̃ | Tr(Fr_{q}, F_{x}) ≠ 0}, and let us define the IC-support to be the finite set ICsupp(F) = {x ∈ W̃ | m(F, x) ≠ 0}. An argument in [GH], §7 shows that supp(M̃_{u,v}) ⊆ {x ∈ W̃ | x ≤ uv} = ICsupp(M̃_{u,v}). Using the Kottwitz conjecture (see end of section 4), we get the analogous result for nearby cycles: supp(RΨ_{μ}) ⊆ Adm(μ) = ICsupp(RΨ_{μ}). For general F, the precise relation between the sets supp(F) and ICsupp(F) is not clear. However, from equation (2.3) below, it is clear that any maximal element of ICsupp(F) belongs to supp(F), and vice-versa. Hence the two sets have the same maximal elements.

Let us recall a definition from [GH]. Fix an integer d. We say F satisfies property (P)_{d} if for all y ∈ W̃ we have the equivalent identities

Tr(Fr_{q}, F_{y}) = ε_{d}ε_{y}q^{d}q_{y}^{−1}\overline{Tr(Fr_{q}, F_{y})}  (2.1)
Tr(Fr_{q}, DF_{y}) = ε_{d}ε_{y}q^{−d}q_{y}^{−1}\overline{Tr(Fr_{q}, DF_{y})},  (2.2)

where ε_{d} = (−1)^{d}, ε_{y} = (−1)^{ℓ(y)}, q_{y} = q^{ℓ(y)}, DF denotes the Verdier dual of F, and h ↦ \overline{h} is the involution of ℤ[q^{1/2}, q^{−1/2}] determined by q^{1/2} ↦ q^{−1/2}. For example, in [GH] we proved that RΨ_{μ} satisfies (P)_{d} with d = ℓ(μ) and M̃_{u,v} satisfies (P)_{d} with d = ℓ(uv).

Lemma 2.1 Let F be any object of P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) and let d be any positive integer. Then for any x ∈ W̃, the following statements are equivalent:

(1) For all w ≥ x, Tr(Fr_{q}, F_{w}) is a polynomial in q having degree ≤ d − ℓ(w);

(2) For all w ≥ x, m(F, w) is a polynomial in q having degree ≤ d − ℓ(w).

If F satisfies property (P)_{d}, then these are also equivalent to

(3) For all w ≥ x, the weights of the stalk complex F(\frac{d}{2})_{w} are ≤ d − ℓ(w).

Proof. For x, y ∈ W̃ let P_{x,y}(q) denote the corresponding Kazhdan-Lusztig polynomial. For any x ∈ W̃ we have using [KL2] the identity

Tr(Fr_{q}, F_{x}) = ε_{x}m(F, x) + ∑_{w∈ICsupp(F)_{w>x}} ε_{w}m(F, w)P_{x,w}(q).  (2.3)

The maximal elements of the sets supp(F) and ICsupp(F) agree, and (2.3) implies the equivalence (1) ⇔ (2) holds for x a maximal element. The case of general x follows by descending induction with respect to ≤, using (2.3) and the bound deg_{q}P_{x,w} < \frac{1}{2}(ℓ(w) − ℓ(x)) if x < w.

Next we will prove that a statement closely related to the implications (3) ⇔ (2) holds without assuming F satisifies property (P)_{d}. We claim that the following statements are equivalent.

(i) For all w ≥ x, the weights of F(\frac{d}{2})_{w} are ≤ d − ℓ(w);

(ii) for all w ≥ x, deg_{q}m(F, w) ≤ d − ℓ(w).

Proof of claim. Let us prove (i) ⇒ (ii). We fix an element w ≥ x and we may assume w ∈ ICsupp(F). The condition in (i) holds for w if and only if for all w′ ≥ w and i ∈ ℤ such that m(F, w′, i) ≠ 0 we have weight(IC_{w′}(\frac{ℓ(w′)}{2})(\frac{d}{2}−\frac{ℓ(w′)}{2}−i)) ≤ d−ℓ(w). Since IC_{w′}(\frac{ℓ(w′)}{2}) is a pure weight zero perverse sheaf, this last inequality holds if and only if 2i−d+ℓ(w′) ≤ d−ℓ(w), i.e., i ≤ d − \frac{ℓ(w′)+ℓ(w)}{2}. Taking w′ = w, we get deg_{q}m(F, w) ≤ d − ℓ(w), as desired.

Now we prove (ii) ⇒ (i). Assume w ≥ x. The hypothesis means that for every w′ ≥ w, and i such that m(F, w′, i) ≠ 0 we have i ≤ d − ℓ(w′) which is a fortiori ≤ d − \frac{ℓ(w)+ℓ(w′)}{2}. Now reversing the steps in the paragraph above, we see this implies that weight(F(\frac{d}{2})_{w}) ≤ d−ℓ(w). This proves the claim.

In particular we see that (1) and (2) each imply (3) with no extra assumptions on F.

It remains to prove that if F satisfies (P)_{d} and also satisfies the equivalent conditions (i) and (ii) above, then for all w ≥ x the functions Tr(Fr_{q}, F_{w}) are polynomials in q having degree ≤ d − ℓ(w). First of all the inequality deg_{q}Tr(Fr_{q}, F_{w}) ≤ d − ℓ(w) follows from (ii) and (2.3). The fact that Tr(Fr_{q}, F_{w}) is actually a polynomial in q follows from this inequality and the identity (2.1) defining property (P)_{d}. □

3 A bound on the weights of nearby cycles on each stratum

By Lemma 2.1, Theorem 1.1 can be reformulated as follows.

Theorem 3.1 For each w ∈ Adm(μ) and each closed point z ∈ B_{w}, the weights of the stalk complex RΨ_{μ}(\frac{ℓ(μ)}{2})_{z} are ≤ ℓ(μ) − ℓ(w).

To prove this we will first prove a weaker statement that holds in much greater generality.

3.1 Bounding weights generically on strata for nearby cycles over Henselian traits

Let (S, s, η) be the spectrum of a complete discrete valuation ring with finite residue field k(s). Let k(\overline{η}) be a separable closure of the generic point k(η), and let \overline{S} be the normalization of S in \overline{η}, with closed point \overline{s}. Let X be an S-variety, i.e. an integral separated scheme, flat and of finite type over S.

Let Xs̄ denote the geometric special fiber, and let D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}) denote the category consisting of objects in the “derived category” D_{c}^{b}(Xs̄, \overline{ℚ}_{ℓ}) of ℓ-adic \overline{ℚ}_{ℓ}-sheaves, equipped with a Weil structure. (See e.g. [GH], §2).

We assume that the special fibre X_{s} of X is stratified by locally closed subschemes X_{α}, and that each X_{α} is equidimensional, of dimension d_{α}, say.

Definition 3.2 We say that F ∈ D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}) generically satisfies the sharp bound with respect to d, if for each stratum X_{α} there exists an open dense subset U_{α} ⊆ X_{α} such that for all z ∈ U_{α}, the weights on the stalk F_{z} are ≤ d − dim X_{α}.

We say that F satisfies the sharp bound with respect to d if for each α we may take U_{α} = X_{α}.

We say that F (generically) satisfies the sharp bound on its weights, if it (generically) satisfies the sharp bound w. r. t. d = dim X_{s}.

For any X/S such that X_{η} is geometrically integral, we let IC(X_{η}) denote the (perverse) intersection complex of X_{η}, and we let ^{0}IC(X_{η}) = IC(X_{η})(\frac{dimX_{η}}{2} ) denote the Tate-twist of IC(X_{η}) that is a self-dual and weight zero perverse sheaf on X_{η}. Finally, we define the “normalized” nearby cycles complex in D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}) by

0RΨ(IC(X_{η})) = RΨ^{X}(^{0}IC(X_{η})).

It is well-known that nearby cycles preserve (middle) perversity, and so ^{0}RΨ(IC(X_{η})) is an object of P_{Weil}(X, \overline{ℚ}_{ℓ}), the full subcategory D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}) whose objects are perverse.

Note that it would be more correct to say that the nearby cycles is an object of the category D_{c}^{b}(X×_{s}η, \overline{ℚ}_{ℓ}), or of its subcategory P (X×_{s}η, \overline{ℚ}_{ℓ}) of perverse objects (see e.g. [GH],§2, 5). But as in loc. cit., by choosing a lift Fr_{q} ∈ Gal(k(η̄)/k(η)) of the geometric Frobenius generator in Gal(k(s̄)/k(s)), we can and will regard the nearby cycles as being an object in P_{Weil}(X, \overline{ℚ}_{ℓ}). This causes no problems in the statement or application of the proposition below.

Proposition 3.3 Let X/S be an S-variety such that X_{η} is geometrically integral. Suppose that the special fiber X_{s} is stratified by locally closed subvarieties X_{α} which are equidimensional of dimension d_{α}. Then ^{0}RΨ(IC(X_{η})) generically satisfies the sharp bound on its weights.

Proof. By the calculations of Rapoport-Zink [RZ], the sharp bound holds in the case where X/S is proper strictly semi-stable, in the terminology of de Jong [deJ]. We will reduce the proposition to that case.

The key ingredient in the reduction is the following result of de Jong ([deJ], Theorem 6.5).

Theorem 3.4 (de Jong) Let X/S be an S-variety. Then there exists a trait S′ = (S′, s′, η′) finite over S, an S′-variety X′ for which there is an alteration X′ → X of S-varieties, and an open immersion j : X′ ↪ \overline{X′} of S′-varieties such that \overline{X′} is a proper strictly semi-stable S′-variety:

\xymatrix{\overline{X′} \ar[d] & X′ \ar[l]_{j} \ar[d] \ar[r] & X \ar[d] \\ S′ & S′ \ar[l]_{id} \ar[r] & S.}

We will make use of this theorem in exactly the same manner as [GH], §10, where we proved that ^{0}RΨ(IC(X_{η})) is mixed.

Let r : X_{S′} → X be the projection morphism defined by the diagram in Theorem 3.4. Note that ^{0}IC(X_{η′}) = ^{0}IC(X_{η})_{η′}. By invariance of nearby cycles under change of traits ([SGA 4 1/2, Thm. finitude 3.7]), it then follows that r^{∗}RΨ^{X}(^{0}IC(X_{η})) = RΨ^{X_{S′}} (^{0}IC(X_{η′})), as objects of P_{Weil}(X_{S′}, \overline{ℚ}_{ℓ}). We have used here that X_{S′} and X have “the same” geometric special fiber Xs̄, since k(s) is finite. Thus, to bound the weights of ^{0}RΨ(IC(X_{η})) generically on each stratum X_{α}, it is enough to do so for RΨ^{X_{S′}} (^{0}IC(X_{η′})) on each stratum X_{α,s′}, the base-change of X_{α} via s′ → s.

We may therefore replace X/S with X_{S′}/S′. In effect, this means we may assume S′ = S and that we have an alteration π : X′ → X of S-varieties and an open immersion j : X′ ↪ \overline{X′} of S-varieties such that \overline{X′}/S is a proper strictly semi-stable S-variety.

Let us denote simply by π the induced morphisms π : X′_{η} → X_{η} and π : X′_{s} → X_{s} on generic and special fibers, and on the corresponding geometric fibers. For any object K of D_{c}^{b}(Xη̄, \overline{ℚ}_{ℓ}) or D_{c}^{b}(Xs̄, \overline{ℚ}_{ℓ}), let ^{p}H^{i}K (resp. H^{i}K) denote the i-th cohomology sheaf of K for the perverse (resp. standard) t-structure. Also, let ^{p}τ^{≥i} be the perverse truncation functor. Let ^{p}π_{∗} := ^{p}H^{0}π_{∗} denote the perverse version of the push-forward functor π_{∗} := Rπ_{∗} on derived categories. Since π is proper, π_{∗} = π_{!} and hence ^{p}π_{∗} = ^{p}π_{!}.

Let Λ_{η}^{X′} denote the constant sheaf \overline{ℚ}_{ℓ} on X′_{η}. For n = dim(X_{η}) let A^{X′} = Λ_{η}^{X′}[n](\frac{n}{2}). The smoothness of X′_{η} means that A^{X′} is a self-dual perverse sheaf on X′_{η} which is pure of weight 0.

In this circumstance, we know by [GH], Lemma 10.7, that ^{0}IC(X_{η}) is a subquotient of the perverse sheaf ^{p}π_{∗}(A^{X′}). By the exactness of the functor RΨ^{X}, the perverse sheaf RΨ^{X}(^{0}IC(X_{η})) is a subquotient of RΨ^{X}(^{p}π_{∗}A^{X′}) = ^{p}π_{∗}RΨ^{X′}(A^{X′}) in the category P_{Weil}(X, \overline{ℚ}_{ℓ}). Thus, it is enough to prove the sharp bound holds for the complex pπ_{!}RΨ^{X′}(A^{X′})_{z}, whenever z ∈ U_{α} for some open dense subset U_{α} ⊆ X_{α}.

To bound the weights of ^{p}π_{!}RΨ^{X′}(A^{X′})_{z}, we claim it is enough to bound the weights of π_{!}RΨ^{X′}(A^{X′})_{z}. To see this, note that π_{!}A^{X′} is a pure complex of “geometric origin” on X_{η}, in the terminology of [BBD], §5, 6. Hence for each i the distinguished triangle

\xymatrix{^{p}H^{i}π_{!}A^{X′} \ar[r] & ^{p}τ^{≥i}π_{!}A^{X′} \ar[r] & ^{p}τ^{≥i+1}π_{!}A^{X′}}

becomes a direct sum over η̄ (loc. cit. Thm. 5.4.5, 6.2.5). Applying the functor RΨ^{X′}(·)_{z} we get a distinguished triangle

\xymatrix{^{p}H^{i}π_{!}RΨ^{X′}(A^{X′})_{z} \ar[r] & ^{p}τ^{≥i}π_{!}RΨ^{X′}(A^{X′})_{z} \ar[r] & ^{p}τ^{≥i+1}π_{!}RΨ^{X′}(A^{X′})_{z}}

which becomes a direct sum once we forget the Galois actions. Thus, for each j we have a short exact sequence of ℓ-adic Galois modules

\xymatrix{0 \ar[r] & H^{j}(^{p}H^{i}K_{z}) \ar[r] & H^{j}(^{p}τ^{≥i}K_{z}) \ar[r] & H^{j}(^{p}τ^{≥i+1}K_{z}) \ar[r] & 0,}

where for brevity we have written K_{z} in place of π_{!}RΨ^{X′}(A^{X′})_{z}. Note that for i << 0 the middle term is H^{j}(π_{!}RΨ^{X′}(A^{X′})_{z}). It now follows easily by ascending induction on i that if the sharp bound generically holds for the weights of π_{!}RΨ^{X′}(A^{X′})_{z} then the sharp bound generically holds for each ^{p}H^{i}π_{!}RΨ^{X′}(A^{X′})_{z}, and in particular for ^{p}π_{!}RΨ^{X′}(A^{X′})_{z}.

Now returning to the diagram in Theorem 3.4, let Λ\overline{X′} ; η denote the constant sheaf \overline{ℚ}_{ℓ} on \overline{X′}_{η}, with corresponding perverse Tate-twist A\overline{X′} := Λ\overline{X′} ; η[n](\frac{n}{2}). Let D_{i} be the irreducible components on the special fiber of X′. The formula

j^{∗}RΨ\overline{X′}(A\overline{X′}) = RΨ^{X′}(A^{X′})

and the results of Rapoport-Zink [RZ] for the proper strictly semi-stable model \overline{X′} → S show that the weights of RΨ^{X′}(A^{X′}) on any intersection ∩_{i∈J}D_{i} are ≤ dim(X′ ; s̄) − dim(∩_{i∈J}D_{i}). Here J denotes any finite subset of the set indexing the components D_{i}.

Now we can complete the proof of the proposition. Since π : X′_{s} → X_{s} is proper, the union of the sets X_{α}∩π(∩_{i∈I}D_{i}), where I ranges over all subsets of indices such that dim(∩_{i∈I}D_{i}) < d_{α}, is a proper closed subset of X_{α}. Moreover, since X_{α} is equidimensional the complement U_{α} ⊆ X_{α} of this closed subset is open and dense. For z ∈ U_{α} the fiber π^{−1}(z) does not meet any intersection of irreducible components in X′_{s} of form ∩_{i∈I}D_{i} whose dimension is < d_{α}. It follows that the weights of RΨ^{X′}(A^{X′}) on the fiber π^{−1}(z) are ≤ dim(X′ ; s̄) − d_{α}, by the above remarks. Hence for z ∈ U_{α} the weights of π_{!}RΨ^{X′}(A^{X′})_{z} = RΓ_{c}(π^{−1}(z), RΨ^{X′}(A^{X′})) are ≤ dim(X′ ; s̄) − d_{α}, and we are done. □

3.2 Application to the affine flag manifold

Of course, the generic sharp bound on the weights is most interesting in cases where the weights are actually constant along the strata, in particular when one is dealing with sheaves equivariant under some group action, where the stratification is given by the orbits. This is the situation we considered in [GH]: in either the p-adic or the function-field setting, the sheaves RΨ_{μ} we consider are B-equivariant hence constant along the B-orbits B_{w}, for w ∈ W̃; see loc. cit. §2 and [HN1],[Ga]. Hence we have the following corollary of Proposition 3.3, which immediately proves Theorems 3.1 and 1.1.

Corollary 3.5 Since RΨ_{μ}(\frac{ℓ(μ)}{2}) is B-equivariant, it satisfies the sharp bound on its weights with respect to d = ℓ(μ) = dim(Q_{μ}) and the stratification by B-orbits.

3.3 The case where RΨ is smooth along the strata

The referee for this article suggested that we include the following consequence of Proposition 3.3. We thank the referee for his remarks.

We say a mixed complex F ∈ D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}) is smooth along a stratum X_{α} if for all i the restriction of the cohomology sheaf H^{i}F is a smooth (= lisse) Weil sheaf on X_{α}.

Proposition 3.6 In the situation of Proposition 3.3, assume that RΨ(IC(X_{η})) is smooth along every stratum X_{α}. Then ^{0}RΨ(IC(X_{η})) satisfies the sharp bound on its weights, with respect to this stratification.

Proof. This follows immediately from Proposition 3.3 and the semicontinuity of weights theorem (e.g. [KW], Theorem I.2.8), applied to the smooth Weil sheaves ^{0}R^{i}Ψ(IC(X_{η}))|_{X_{α}}. □

We remark that this proposition does not allow us to avoid invoking that RΨ_{μ} is B-equivariant in the proof of Corollary 3.5, because a priori we know only that RΨ_{μ} is constructible along the B-orbits. We could of course replace the stratification by B-orbits with a finer stratification along whose strata RΨ_{μ} definitely is smooth. But since Proposition 3.6 would pertain only to the finer stratification, the end result could be weaker than what we assert in Corollary 3.5.

4 Bounding degrees of coefficients in Wakimoto functions

In this section we give a combinatorial result which proves Theorem 1.2 and which also yields —via the Kottwitz conjecture (4.1)— a different proof of Theorem 1.1.

By Lemma 2.1, Theorem 1.2 can be reformulated as follows.

Theorem 4.1 For each w ≤ uv, the function Tr(Fr_{q}, (M̃_{u,v})_{w}) is a polynomial in q having degree ≤ ℓ(uv) − ℓ(w).

We will need some further notation. Recall we consider a split connected reductive group G over a field k. Let us fix a choice of Borel subgroup B ⊂ G and maximal torus T ⊂ B. Let B ⊂ G(k[[t]]) be the Iwahori subgroup whose “reduction modulo t” is B. The data (G, B, T) determine a (based) root system and a set of simple reflections S generating the Coxeter group (W, S), where W = N_{G}T/T is the finite Weyl group. Let w_{0} denote the unique longest element in W. Let us choose an origin in the apartment corresponding to T and a Weyl chamber Ch(B) corresponding to B. Let A denote the base alcove, that is, the unique alcove in the given apartment whose closure contains the origin and which lies in Ch(B). Let B̄ be the unique opposite Borel subgroup, that is, the one such that B ∩ B̄ = T.

Let W_{aff} denote the affine Weyl group of G. It is a Coxeter group with generators S_{aff}, the set of simple reflections through the walls of the opposite alcove w_{0}(A) of the base alcove A: if G is almost simple with B-positive simple roots α_{i} (1 ≤ i ≤ l) and B-highest root α̃, then w(A_{0}) is the alcove

w_{0}(A) = {x ∈ X_{∗}(T) ⊗ ℝ | 0 < ⟨−α_{i}, x⟩ < 1 for all i, and ⟨−α̃, x⟩ < 1}.

In this case the simple affine reflections are the reflections s_{−α_{i}} = s_{α_{i}} and s_{0} := s_{α̃+1} = t_{−α^{∨}}sα̃. (See also [GH], section 2.2.)

The alcove fixed by B is the opposite alcove w_{0}(A). This results from our convention that we embed W̃ into G(k((t))) using the identification of ν ∈ X_{∗}(T) with ν(t) ∈ G(k((t))).

The B̄-positive affine roots α + k are those taking positive values on w_{0}(A). Equivalently, α + k is B̄-positive if and only if either k ≥ 1, or k = 0 and α is B̄-positive.

The set of affine roots carries a left action by the group W̃ by the rule

x · (α + k) = (α + k) ∘ x^{−1}.

Lemma 4.2 Let x ∈ W̃ and suppose α + k is a B̄-positive affine root, with corresponding reflection s = s_{α+k}. Let ≤ denote the Bruhat order on W̃ defined using S_{aff}. Then

(a) x < xs if and only if x · (α + k) is B̄-positive;

(b) x < sx if and only if x^{−1} · (α + k) is B̄-positive.

Proof. It is enough to prove (b). Write A^{−} := w_{0}(A). We note that x < sx if and only if xA^{−} and A^{−} lie on the same side of the hyperplane α + k = 0, which happens if and only if (α + k) ∘ x takes positive values on A^{−}, i.e., x^{−1} · (α + k) is B̄-positive. □

We will also need one more lemma.

Lemma 4.3 Suppose t = s_{β}, where β is a B̄-positive affine root. Let u, v ∈ W̃ and suppose u < ut. Then

uv < utv ⟺ v < tv.

Proof. Note first that utu^{−1} = s_{u·β} and u · β is B̄-positive, by Lemma 4.2. Then we have

uv < utv ⟺ uv < s_{u·β}uv
⟺ (uv)^{−1} · (u · β) is B̄-positive
⟺ v^{−1} · β is B̄-positive
⟺ v < tv,

again by Lemma 4.2. □

For elements u, v ∈ W̃, we consider the Wakimoto functions T̃_{u}T̃_{v^{−1}}^{−1}, see [GH], section 7. Here for x ∈ W̃, the symbol T_{x} denotes the standard generator char(BxB) in the Iwahori-Hecke algbra C_{c}(B\G(𝔽_{q}((t)))/B) and T̃_{x} := q_{x}^{−1/2}T_{x}. Our goal is to prove the following improved version of loc. cit. Proposition 7.6 (where the same conclusion is obtained under the additional hypothesis that T̃_{u}T̃_{v^{−1}}^{−1} has a minimal expression). The indeterminate Q that appears here is defined by the relation q^{1/2}Q = 1 − q.

Proposition 4.4 If T̃_{u}T̃_{v^{−1}}^{−1} = ∑_{x} R_{x,v}^{u}(Q) T̃_{x}, then R_{x,v}^{u}(Q) is a polynomial in Q with integer coefficients, and

deg_{Q}R_{x,v}^{u}(Q) ≤ ℓ(uv) − ℓ(x).

Proof. We may assume u, v ∈ W_{aff}. Consider reduced expressions u = t_{1} ⋯ t_{k} and v = s_{1} ⋯ s_{r}, where t_{j}, s_{i} belong to S_{aff}. Write t̃_{j} resp. s̃_{i}^{−1} as short-hand for T̃_{t_{j}} resp. T̃_{s_{i}}^{−1}. We will prove the desired bound

deg_{Q} t̃_{1} ⋯ t̃_{k} · s̃_{1}^{−1} ⋯ s̃_{r}^{−1}(x) ≤ ℓ(uv) − ℓ(x),

by induction on k, the case k = 0 being obvious. We assume the bound holds for k ≥ 0 and deduce it for k + 1.

Let t = s_{β}, where β is a B̄-positive simple affine root, and suppose u < ut = t_{1} ⋯ t_{k}t.

Case 1: We have v < tv (so by Lemma 4.3, uv < utv). Then because t̃^{−1} = t̃+ Q, we have

t̃_{1} ⋯ t̃_{k}t̃s̃_{1}^{−1} ⋯ s̃_{r}^{−1}(x) = t̃_{1} ⋯ t̃_{k}t̃^{−1}s̃_{1}^{−1} ⋯ s̃_{r}^{−1}(x) − Qt̃_{1} ⋯ t̃_{k}s̃_{1}^{−1} ⋯ s̃_{r}^{−1}(x).

Now by induction the deg_{Q} of the first term on the right is bounded by ℓ(utv)−ℓ(x). Similarly the deg_{Q} of the second term is bounded by 1 + ℓ(uv) − ℓ(x) ≤ ℓ(utv) − ℓ(x).

Case 2: We have tv < v (so by Lemma 4.3, utv < uv). Then we can write a reduced expression for v in the form v = ts_{2} ⋯ s_{r}, and thus we are considering

t̃_{1} ⋯ t̃_{k}t̃t̃^{−1}s̃_{2}^{−1} ⋯ s̃_{r}^{−1}(x) = t̃_{1} ⋯ t̃_{k}s̃_{2}^{−1} ⋯ s̃_{r}^{−1}(x)

and clearly our induction hypothesis implies that the deg_{Q} is bounded by ℓ(utv) − ℓ(x), as desired.

The fact that R_{x,v}^{u} is a polynomial in Q with integer coefficients can also be proved by induction on k = ℓ(u) in a similiar way, or by using the Iwahori-Matsumoto relations in the Hecke algebra (see [H01]). □

Let us now state explicitly what Proposition 4.4 says about the weights of the Wakimoto sheaves M̃_{u,v}. Recall that for u, v ∈ W̃, the Wakimoto sheaf is defined as

M_{u,v} = j_{u!}\overline{ℚ}_{ℓ}[ℓ(u)] ⋆ j_{v∗}\overline{ℚ}_{ℓ}[ℓ(v)],

where j_{u}, j_{v} are the inclusions of the Schubert cells corresponding to u, v into the affine flag variety, where j_{u!}, j_{v∗} denote the derived functors, and where ⋆ is the convolution product of perverse sheaves. The function associated to M_{u,v} via the sheaf-function dictionary is

ε_{u}ε_{v}q_{v}T_{u}T_{v^{−1}}^{−1} = ε_{u}ε_{v}q_{u}^{1/2}q_{v}^{1/2} ∑_{x} q_{x}^{−1/2}R_{x,v}^{u}T_{x},

and so the function associated to M̃_{u,v} is ε_{u}ε_{v}q_{uv}^{1/2} ∑_{x} q_{x}^{−1/2}R_{x,v}^{u}T_{x}.

From [GH], §7 we know that for any power Q^{i} which appears with non-zero coefficient in R_{x,v}^{u}(Q), the integer i has the same parity as ℓ(uv) − ℓ(x). This together with the identity q^{1/2}Q = 1−q and the bound deg_{Q}R_{x,v}^{u}(Q) ≤ ℓ(uv)−ℓ(x) of Proposition 4.4 gives the following result, which is a reformulation of Theorems 4.1 and 1.2.

Corollary 4.5 Let u, v, x ∈ W̃. Then Tr(Fr_{q}, (M̃_{u,v})_{x}) is a polynomial in q of degree ≤ ℓ(uv) − ℓ(x). The sheaf M̃_{u,v}(\frac{ℓ(uv)}{2}) satisfies the sharp bound with respect to d = ℓ(uv) and the stratification by B-orbits.

We remark that via the Kottwitz conjecture stated below, this corollary can be used to give another proof of Theorem 1.1. The Kottwitz conjecture is the following identity in the Iwahori-Hecke algebra C_{c}(B\G(𝔽_{q}((t)))/B). In the function-field setting it was proved by Gaitsgory [Ga], whose ideas were adapted to prove it in the p-adic setting in [HN1].

Tr(Fr_{q}, RΨ_{μ}) = ε_{μ}q_{μ}^{1/2} ∑_{λ∈Ω(μ)} m_{μ}(λ)Θ_{λ}.  (4.1)

The original form concerned the semi-simple trace of Frobenius, but by virtue of the unipotence of the inertia action on RΨ_{μ} which was proved by Gaitsgory [Ga] (see also [GH] §5), we may replace the semi-simple trace Tr^{ss} with the usual trace, once we have chosen a lift Fr_{q} of geometric Frobenius, as we have already done. See the discussion of this point in [GH], §5.

Let us explain the notation on the right hand side. We let Ω(μ) denote the set of coweights λ ∈ X_{∗}(T) such that for every w ∈ W, μ − wλ is a sum of B-positive coroots. For such a coweight λ, write it as λ = λ_{1} − λ_{2}, where λ_{i} is B-dominant for i = 1, 2. Then following Bernstein we define

Θ_{λ} = T̃_{t_{λ_{1}}} T̃_{t_{λ_{2}}}^{−1}.

Let G^{∨} be the dual group for G. The number m_{μ}(λ) is the multiplicity with which the weight λ appears in the character E_{μ} of the simple G^{∨}-module with highest weight μ.

Note that ℓ(λ) ≤ ℓ(μ) and the difference ℓ(μ) − ℓ(λ) is an even integer. It is now clear that Theorem 1.1 can be deduced, via Lemma 2.1, by applying Corollary 4.5 with u = t_{λ_{1}} and v = t_{−λ_{2}} for all λ ∈ Ω(μ) and by using (4.1).

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Ulrich Görtz: Mathematisches Institut der Universität Bonn, Beringstr. 6, 53115 Bonn, Germany. Email: ugoertz@math.uni-bonn.de

Thomas J. Haines: Mathematics Department, University of Maryland, College Park, MD 20742-4015, USA. Email: tjh@math.umd.edu