The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties

Ulrich Görtz and Thomas J. Haines ^{∗}

Abstract

We study the Jordan-Hölder series for nearby cycles on certain Shimura varieties and Rapoport-Zink local models, and on finite-dimensional pieces of Beilinson’s deformation of the affine Grassmannian to the affine flag variety (and their p-adic analogues). We give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups, and we also provide an algorithm to compute multiplicities, in terms of the affine Hecke algebra.

MSC-Classification: 14M15, 14G35, 20C08

1 Introduction

This article has several aims. The initial motivation was our desire to understand, as explicitly as possible, the irreducible constituents of the nearby cycles sheaf RΨ on the reduction modulo p of a Shimura variety with Iwahori-level structure at a fixed prime p. In some sense, the complexity of RΨ is a measure of the singularities in the reduction modulo p. Moreover, we were motivated by the role RΨ plays in the computation of the semi-simple local zeta function at p for such a Shimura variety, cf. [RZ1], [HN1].

Via the relationship between the Shimura variety and its Rapoport-Zink local model [RZ2], one can translate the problem into that of understanding the nearby cycles on a local model. The latter can be embedded as a finitedimensional piece of 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 p-adic group G (M exists at least if G is either GL_{n} or GSp_{2n}; see [HN1] and section 8). A very similar deformation Fl_{X} over a smooth curve X (due to Beilinson) exists for any group G in the function field setting, and has been extensively studied by Gaitsgory [Ga].

The “maximal parahoric” subgroup G(O) in the loop group of G acts on Grass, the generic fiber of the deformation M or Fl_{X}. Fixing a dominant coweight μ of G, one may therefore consider nearby cycles RΨ = RΨ(IC_{μ}), taken with respect to the deformation, where IC_{μ} is the intersection complex on the closure \overline{Q}_{μ} of the G(O)-orbit in Grass indexed by μ. Letting B denote the standard Iwahori subgroup of G(O), one may show that RΨ is a B-equivariant perverse sheaf on the affine flag variety, with certain additional properties (e.g., it belongs to the category P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) of section 4). Thus, quite generally, we were led to the problem of understanding the Jordan-Hölder series for objects F of the category P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}). It turns out that the only irreducible objects in this category are Tate-twists of intersection complexes on finite-dimensional Schubert varieties B\overline{w} in Fl (they are indexed by elements w in the extended affine Weyl group W̃ for G; we denote such intersection complexes by IC_{w}). Thus, we may define non-negative “multiplicities” m(F, w, i) by the identity in P_{q}^{B}(Fl, \overline{ℚ}_{ℓ})

∗Research supported in part by NSF grant DMS 0303605

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

Our main theorem is the following cohomological interpretation for the integers m(F, w, i) for sheaves F which satisfy a somewhat technical hypothesis (Property (P) of section 4.3). This property holds for all the nearby cycles we consider, and also for a much larger class of objects (those with a suitable filtration by Wakimoto sheaves; see section 7).

Theorem 1.1 Suppose F ∈ P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) satisfies property (P) for the integer d. Then for each w ∈ W̃,

∑_{i} m(F, w, i)q^{i} = (−q)^{d}Tr(Fr_{q}, H_{c}^{•}(Fl, DF ⊗ IC(B\overline{w}))).

Here DF denotes the Verdier dual of F; there is also a “dual” formulation (see Theorem 4.7) in which F appears rather than DF. We introduce and study the intersection complex IC(B\overline{w}) appearing here in section 3. In a certain sense, it is the intersection complex for an orbit B^{w} for the “opposite” Iwahori subgroup B^{−}. Such orbits are infinite-dimensional, and so it is necessary to construct IC(B\overline{w}) by pull-back from certain finite-dimensional quotients (introduced by Faltings [F]) of open sets Ω ⊂ Fl. The basic properties of IC(B\overline{w}) play a key role in the proof. The other main ingredient is the cohomological interpretation of inverse Kazhdan-Lusztig polynomials, announced in [KL2]. In fact, the Kazhdan-Lusztig theorem can be seen as a special case of our main theorem (see section 4.5). Since no proof of the Kazhdan-Lusztig theorem in exactly our setting has appeared in the literature, we provide a proof for this result in section 3. Here again we make essential use of our sheaf IC(B\overline{w}) and of the quotients of Faltings [F] to pass from ind-schemes to ordinary finite-dimensional schemes.

The main theorem was discovered in the attempt to find a conceptual explanation for the following remarkable fact (Cor. 6.3): Let M_{μ} denote the schemetheoretic closure in our deformation of \overline{Q}_{μ}. Let τ denote the element of W̃ indexing the unique dimension zero Iwahori-orbit in the special fiber of M_{μ} (i.e., the “most singular point”). Then we have

Corollary 1.2 For RΨ = RΨ(IC_{μ}), the multiplicity m(RΨ, τ, i) is the 2i-th intersection Betti number for \overline{Q}_{μ}.

This fact, noticed after several computer-aided calculations using the algorithm described below, gave us the first hint that multiplicity functions have a cohomological interpretation.

It is clear that knowing the multiplicities m(F, w, i) for every w and every i is equivalent to knowing an explicit formula for the trace function Tr(Fr_{q}, F) as an element in the Iwahori-Hecke algebra (see section 4.2). On the other hand, the Kottwitz conjecture (proved in [Ga] and [HN1], cf. section 2.7) identifies the semi-simple trace function Tr^{ss}(Fr_{q}, RΨ(IC_{μ})) as an explicit sum of functions Θ_{λ}, each of which can be computed (with a computer program, see sections 7,9). Thus, there is an algorithm to compute the multiplicities m(RΨ, w, i) in any given case. (Actually, to justify the algorithm, one has to know that Tr^{ss}(Fr_{q}, RΨ) = Tr(Fr_{q}, RΨ); see comments below.) Similarly, the algorithm works to compute the numbers m(F, w, i) whenever F has a suitable filtration by Wakimoto sheaves, cf. section 7. In section 9, we give the results of several such computer-aided calculations, and we give explanations for some empirical observations we made.

Another consequence of the Kottwitz conjecture is that the functions Tr^{ss}(Fr_{q}, RΨ(IC_{μ})) form a basis for the center Z(H) of the Iwahori-Hecke algebra for G, as μ ranges over dominant coweights. In terms of the standard generators T_{x} = char(BxB) (x ∈ W̃) for H, we have the identity

Tr^{ss}(Fr_{q}, RΨ) = ∑_{x} (∑_{w≥x} (∑_{i} m(RΨ, w, i)q^{i} ε_{w}P_{x,w})) T_{x},

where ≤ is the Bruhat order on W̃ and P_{x,w} are the Kazhdan-Lusztig polynomials [KL1], cf. section 4.2. Our results on m(RΨ, w, i) therefore give us both a conceptual and an algorithmic way to make this formula explicit, and this works for any group G. Other approaches to computing elements in Z(H) explicitly in terms of the T_{x}-basis were given previously in [H1],[H3], [HP], and [S].

Other aims of the paper were to provide proofs of several necessary results of a foundational nature, some of which are very general and might be useful in other contexts. Here is a list of the highlights:

• A study of categories of B-equivariant perverse sheaves on Fl; section 4;

• For finite-type schemes X over a trait (S, s, η), the canonical decomposition of the category P (X ×_{s} η, \overline{ℚ}_{ℓ}) of middle perverse sheaves on Xs̄ endowed with a compatible continuous action of Gal(η̄/η), into “unipotent” and “nonunipotent” subcategories; section 5; • The proof of Gaitsgory’s theorem that our sheaves RΨ are unipotent (in the above sense), and the consequence that Tr^{ss}(Fr_{q}, RΨ) = Tr(Fr_{q}, RΨ), alluded to above; section 5;

• If X is finite-type over S, an unequal characteristic Henselian trait, and X_{η} is smooth, then the nearby cycles sheaf RΨ^{X}(\overline{ℚ}_{ℓ}) is mixed; section 10;

• If X as above is proper, the existence of the weight spectral sequence; section 8.

Returning now to the situation of Shimura varieties Sh considered in section 8, the Shimura datum (G, X) provides us with a minuscule coweight μ of G_{ℚ_{p}}, and the model M_{μ} above coincides with the Rapoport-Zink local model M^{loc}, so the above results apply to give the Jordan-Hölder series for RΨ_{Sh}. The irreducible constituents are Tate-twists of intersection complexes IC_{Sh,w} relative to a stratification ∐ Sh_{w} on the special fiber Sh_{𝔽_{p}}, induced by the B-orbit stratification on M_{μ} (see [GN]), indexed by the subset Adm(μ) of section 2.4 ^{1}. If Sh is proper over the ring of integers O_{E} of the reflex field E, then the weight spectral sequence _{W} E^{•,•} ⇒ H^{•}(Sh_{\overline{ℚ}_{p}}, \overline{ℚ}_{ℓ}[ℓ(μ)]) exists (by virtue of the Appendix) and its E_{1}-term can be made somewhat explicit. We have the following result (see section 8 for details).

Theorem 1.3 (a) In the category P (Sh ×_{s} η, \overline{ℚ}_{ℓ}), we have

RΨ_{Sh}^{ss}(\overline{ℚ}_{ℓ}[ℓ(μ)]) = ⨁_{w∈Adm(μ)} ⨁_{i=0}^{ℓ(μ)−ℓ(w)} IC_{Sh,w}(−i)^{⊕m(RΨ_{Mμ},w,i)}.

(b) Assume Sh is proper over O_{E}. Then there is an isomorphism of \overline{ℚ}_{ℓ}-spaces

W E_{1}^{pq} = ⨁_{{w∈Adm(μ) \atop ℓ(w)+2j=−p}} IH^{p+q+ℓ(w)}(\overline{Sh}_{w}, \overline{ℚ}_{ℓ})^{⊕m(RΨ_{Mμ},w,j)}.

The multiplicity functions ∑_{i} m(RΨ(IC_{μ}), w, i)q^{i} appear to carry fundamental information. In all the cases we computed, they are actually polynomials in q of degree exactly ℓ(μ) − ℓ(w) (and they vanish if w ∉ Adm(μ)). It is perhaps striking that we do not know how to prove the polynomial nature of the multiplicity function in general. It can be proved when μ is either minuscule or a coweight for GL_{n}, by using the existence of minimal expressions for the functions Θ_{λ} that arise in the Kottwitz conjecture, see section 9. We provide a possible geometrical approach in Part II of the Appendix (section 10).^{2}

1Recall that Sh is usually geometrically disconnected. However the various connected components carry the “same” stratification, induced in each case by that on M_{μ}. To be precise, in all statements the symbol IC_{Sh,w} should be interpreted as the direct sum, over all geometric connected components, of the intersection complexes indexed by w on the various components.

In particular this applies to the results in section 8.

2After this article was submitted, we proved in the most general case that the multiplicity functions are polynomials in q having degree at most ℓ(μ) − ℓ(w), see [GH].

2 Notation and preliminaries

2.1 Galois structures on derived categories

As usual, p will denote a fixed prime number, and q will always denote a power of p. We will work over Henselian traits (S, s, η) of residue characteristic p. These will usually be of the form S = Spec(𝔽_{p}[[t]]) (the function field setting), or S = Spec(ℤ_{p}) (the p-adic setting). We choose a separable closure η̄ of η and define the Galois group Γ = Gal(η̄/η) and the inertia subgroup Γ_{0} = ker[Gal(η̄/η) → Gal(s̄/s)], where s̄ is the residue field of the normalization S̄ of S in η̄.

Suppose the residue field k(s) has cardinality q. Any element of Γ which lifts the inverse Frob_{q} of the topological generator a ↦ a^{q} of Gal(s̄/s) will be called a geometric Frobenius and will be denoted by Fr_{q}. We will often use the same symbol Fr_{q} to denote the Frobenius automorphism id_{X_{0}} × Frob_{q} : X_{0} × Spec(\overline{𝔽}_{q}) → X_{0} × Spec(\overline{𝔽}_{q}) for an 𝔽_{q}-scheme X_{0}. For the most part, we leave it to context to dictate what is meant by the symbol Fr_{q} in each situation.

Let X be a scheme of finite type over a finite (or algebraically closed) field k. (The following also works if we assume that k is the fraction field of a discrete valuation ring R with finite residue field, and that X is finite-type over R, cf. [Ma].) Denote by \overline{k} an algebraic closure of k, and by X\overline{k} the base change.

We denote by D_{c}^{b}(X, \overline{ℚ}_{ℓ}) the ’derived’ category of \overline{ℚ}_{ℓ}-sheaves on X. Note that this is not actually the derived category of the category of \overline{ℚ}_{ℓ}-sheaves, but is defined via a limit process. See [BBD] 2.2.14 or [Weil2] 1.1.2 for more details; see also section 5. Nevertheless, D_{c}^{b}(X, \overline{ℚ}_{ℓ}) is a triangulated category which admits the usual functorial formalism, and which can be equipped with a ’natural’ t-structure having as its core the category of \overline{ℚ}_{ℓ}-sheaves. If f : X ⟶ Y is a morphism of schemes of finite type over k, we denote by f_{∗}, f_{!} the derived functors D_{c}^{b}(X, \overline{ℚ}_{ℓ}) ⟶ D_{c}^{b}(Y, \overline{ℚ}_{ℓ}).

We will denote by P (X, \overline{ℚ}_{ℓ}) the full subcategory of D_{c}^{b}(X, \overline{ℚ}_{ℓ}) consisting of middle perverse sheaves. See [BBD] or [KW] for a detailed discussion of this notion.

Now let X be a scheme of finite type over s. The absolute Galois group Gal(\overline{η}/η) acts on Xs̄ through its quotient Gal(\overline{s}/s).

Definition 2.1 (cf. [SGA 7], exp. XIII)

(1) The category D_{c}^{b}(X ×_{s} η, \overline{ℚ}_{ℓ}) is the category of sheaves F ∈ D_{c}^{b}(Xs̄, \overline{ℚ}_{ℓ}) together with a continuous action of Gal(\overline{η}/η) which is compatible with the action on Xs̄. (Continuity is tested on cohomology sheaves: see section 5.)

(2) Similarly, P (X ×_{s} η, \overline{ℚ}_{ℓ}) denotes the category of perverse sheaves F in P (Xs̄, \overline{ℚ}_{ℓ}) together with a continuous action of Gal(\overline{η}/η) which is compatible with the action on Xs̄.

Analogously, assuming k(s) has cardinality q, we let D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}) denote the category of pairs (F, F_{q}^{∗}) where F belongs to D_{c}^{b}(Xs̄, \overline{ℚ}_{ℓ}), and F_{q}^{∗} : Fr_{q}^{∗}F ˜→ F is an isomorphism in that category. The definition of P_{Weil}(X, \overline{ℚ}_{ℓ}) is similar.

Any choice of geometric Frobenius Fr_{q} ∈ Γ gives rise naturally to a functor

D_{c}^{b}(X ×_{s} η, \overline{ℚ}_{ℓ}) ⟶ D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}).

This induces an analogous functor on the categories of perverse sheaves.

We also have a natural functor

P (X_{s}, \overline{ℚ}_{ℓ}) ⟶ P_{Weil}(X, \overline{ℚ}_{ℓ}), F_{0} ↦ (F, F_{q}^{∗}),

which is a full embedding, and its essential image is a subcategory which is stable under extensions and subquotients; see [BBD], Prop. 5.1.2. In particular, a Weil perverse sheaf which has a filtration such that the graded pieces are all defined over s, lies in the essential image of this functor.

2.2 The affine Grassmannian and flag variety

We briefly recall the definition of the affine Grassmannian Grass, and the affine flag variety Fl.

We fix a field k and suppose G is a split connected reductive group over k. Fix a maximal torus T and a Borel subgroup B containing T. Let Λ_{+} ⊂ X_{∗}(T) denote the set of B-dominant integral coweights for G. By W we denote the finite Weyl group N_{G}(T)/T, by W_{aff} the affine Weyl group, and by W̃ the extended affine Weyl group N_{G}(T)/T_{O}, where O = k[[t]].

Consider G(k((t))) as an ind-scheme over k. Denote by B ⊂ G(k((t))) the standard Iwahori subgroup, i. e. the inverse image of B under the projection G(k[[t]]) ⟶ G(k).

The affine Grassmannian Grass (over the field k) is the quotient (of fpqcsheaves) G(k((t)))/G(k[[t]]); it is an ind-scheme. If G = GL_{n} and R is a kalgebra, Grass(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]]^{×}. In the same way, one can construct the affine Grassmannian over ℤ. By the Cartan decomposition we have a stratification into G(k[[t]])-orbits:

Grass = ∐_{μ∈Λ_{+}} 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 if μ − λ is a sum of B-positive coroots.

The affine flag variety Fl over k is the quotient G(k((t)))/B; it is an ind-scheme, too. As before, we have a modular interpretation: e.g., for GL_{n}, Fl(R) is the space of complete lattice chains in R((t))^{n}. The Iwahori group B acts on Fl, and we get a decomposition

Fl = ∐_{w∈W̃} BwB/B.

Here we embed W̃ = X_{∗}(T) ⋊ W into G(k((t))) by using the aforementioned embedding of X_{∗}(T), and by choosing a representative for w ∈ W = N_{G}(T)/T in the group N_{G}(T)∩G(k[[t]]). We write B_{w} = BwB/B; B_{w} is called the Schubert cell associated to w. Its closure, which we denote by B\overline{w}, is a finite-dimensional projective variety, and is called a Schubert variety. We have

B\overline{w} = ∐_{v≤w} B_{v}.

Here ≤ denotes the Bruhat order on W̃ determined by the affine reflections S_{aff} through the walls of the opposite base alcove, i.e., the alcove w_{0}(A), where w_{0} is the longest element in W and A ⊂ X_{∗}(T) ⊗ ℝ is the unique alcove in the Bdominant chamber whose closure contains the origin. With our normalization of the embedding of X_{∗}(T) into G(k((t))), the alcove fixed by the Iwahori B is w_{0}(A), see [IM] or [HKP].

The length function ℓ: W̃ → ℤ_{≥0} used thoughout this paper is always defined with respect to the Coxeter system (W_{aff}, S_{aff}). Thus, B_{w} is an affine space isomorphic to 𝔸^{ℓ(w)}. Also, we will often write ℓ(μ) instead of ℓ(t_{μ}), where t_{μ} is the translation element in W̃ corresponding to μ ∈ X_{∗}(T).

The analogue of the unipotent radical N^{−} of the opposite Borel B^{−} in the finite-dimensional case, is the subgroup N^{−} ⊂ G(k((t))) which is by definition the inverse image of N^{−} under the projection G(k[t^{−1}]) ⟶ G(k). We can then decompose Fl into N^{−} orbits:

Fl = ∐_{w∈W̃} N^{−}wB/B.

We write B^{w} = N^{−}wB/B. This is an ind-scheme of ’finite codimension’. More precisely, B^{w} ∩ B_{v} ≠ ∅ if and only if w ≤ v, and in this case the intersection has dimension ℓ(v) − ℓ(w); cf. [F]. We denote the closure of B^{w} by B\overline{w}; then

B\overline{w} = ∐_{v≥w} B^{v}.

2.3 Intersection complexes

Let X be a finite-type reduced and geometrically irreducible scheme over a finite field k (or, we may assume k is algebraically closed, or that X is finitetype over a discrete valuation ring with finite residue field and with fraction field k). For any open immersion j : U → X of a smooth irreducible dense open subset, we define IC(X) = j_{!∗}(\overline{ℚ}_{ℓ}), where j_{!∗} is the Goresky-MacPherson-Deligne extension functor (for the middle perversity), cf. [GM], [BBD]. The shift IC(X)[dim(X)] is an irreducible perverse sheaf on X, independent of the choice of U. If X is not reduced, we set IC(X) := IC(X_{red}).

The intersection cohomology groups of X are given by IH^{i}(X) = H^{i}(X, IC(X)) (similarly for compact supports).

We fix a choice of \sqrt{q} ∈ \overline{ℚ}_{ℓ}, needed to define Tate-twists on the categories D_{c}^{b,Weil}(X, \overline{ℚ}_{ℓ}) and P_{Weil}(X, \overline{ℚ}_{ℓ}). It is known that IC(X)[dim(X)](\frac{dim(X)}{2}) is a self-dual pure perverse sheaf of weight zero.

For w ∈ W̃ we have the locally closed immersion j_{w} : B_{w} ↪ Fl. We define the perverse sheaf IC_{w} = IC(Bw̄)[ℓ(w)]. For μ ∈ Λ_{+}, the perverse sheaf IC_{μ} on Grass has a similar definition. Note that there is no Tate-twist in these definitions. This turns out to be the most suitable normalization for our purposes.

2.4 Deformations of the affine Grassmannian to the flag variety

Suppose G = GL_{n} or GSp_{2n}, and S = Spec(ℤ_{p}). There is a deformation from Grass_{ℚ_{p}} to Fl_{𝔽_{p}}, i.e. an ind-scheme M over ℤ_{p} with generic fibre Grass_{ℚ_{p}} and special fibre Fl_{𝔽_{p}}. The second author and Ngô [HN1] defined such a deformation as a union

M = ⋃_{μ∈Λ_{+}} M_{μ},

where M_{μ} is a finite-dimensional projective ℤ_{p}-scheme with generic fiber \overline{Q}_{μ,ℚ_{p}} and special fiber

∐_{w∈Adm(μ)} B_{w,𝔽_{p}}.

Here Adm(μ) is by definition the following finite subset of W̃:

Adm(μ) = {x ∈ W̃ | x ≤ t_{λ}, for some λ ∈ W μ};

we refer to [HN2] for further information about this subset. The model M_{μ} is defined in terms of lattice chains, so this construction works only for G = GL_{n} or GSp_{2n}. It is conceivable, however, that other groups can be handled by using a Plücker description of Grass and Fl, cf. [H2].

In the function field case, suppose G is an arbitrary connected reductive group over k = 𝔽_{p} and suppose X is a smooth curve over k. Then a deformation Fl_{X} has been constructed by Beilinson; see [Ga]. For a distinguished point x_{0} ∈ X, Fl_{X} is an ind-scheme over X whose fiber over x ≠ x_{0} is isomorphic to the product Grass_{k} × G/B, and whose fiber over x_{0} is isomorphic to the affine flag variety Fl_{k}. This deformation also has a modular interpretation, this time in terms of G-bundles on X. We get a deformation over S = Spec(k[[t]]) by base-changing with a formal neighborhood of x_{0} ∈ X.

In either the function-field or the p-adic setting, let Grass denote the “constant” deformation of the affine Grassmannian, i.e., we consider the base change Grass_{ℤ} × S. The modular interpretations of M and Fl_{X} allow us to define a projective morphism of ind-schemes

π : M → Grass

over S (similarly for Fl_{X}). In the GL_{n} case for example, π sends a lattice chain L_{0} ⊂ L_{1} ⊂ ⋯ in M to the lattice L_{0} in Grass. Often, we will use the same letter π to denote the morphism on the special fibers, where it is just the usual projection of ind-schemes G(k((t)))/B → G(k((t)))/G(k[[t]]).

In the sequel, when we use the symbol Fl to denote the affine flag variety, we are often thinking of it as the special fiber of one of the deformations above. Occasionally, we even write Fl for the deformation itself.

2.5 Nearby cycles

Let X denote a finite-type scheme over S. For F ∈ D_{c}^{b}(X_{η}, \overline{ℚ}_{ℓ}), we define the nearby cycles sheaf to be the object in D_{c}^{b}(X ×_{s} η, \overline{ℚ}_{ℓ}) given by

RΨ^{X}(F) = ī^{∗}Rj̄_{∗}(Fη̄),

where ī : Xs̄ ↪ XS̄ and j̄ : Xη̄ ↪ XS̄ are the closed and open immersions of the geometric special and generic fibers of X/S, and Fη̄ is the pull-back of F to Xη̄. It is known that the functor RΨ^{X} preserves perversity, cf. [I].

2.6 Equivariant sheaves

Consider the affine flag variety Fl over 𝔽_{p}. By definition, a (mixed) perverse sheaf on Fl is a (mixed) perverse sheaf which is supported on some finitedimensional part.

Definition 2.2 Denote by a: B × Fl ⟶ Fl the action of the Iwahori group, by i : Fl ⟶ B × Fl the zero section, and by p_{2} : B × Fl ⟶ Fl the projection. A perverse sheaf F is B-equivariant if there exists an isomorphism φ : a^{∗}F ⟶ p_{2}^{∗}F, such that

i) φ is rigidified along the zero section: i^{∗}φ = id_{F},

ii) φ satisfies the cocycle condition: (m × id_{Fl})^{∗}(φ) = p_{23}^{∗}(φ) ∘ (id_{B} ×a)^{∗}(φ) on B × B × Fl, where m: B × B ⟶ B is the multiplication map (cf. [KW] III.15).

Note that since by definition all perverse sheaves on Fl are supported on finitedimensional subschemes of the affine flag variety, and since the Iwahori subgroup acts through a finite quotient on these subschemes (if they are suitably chosen), the fact that we deal with ind-schemes does not pose any problems at this point.

Remark. As Kiehl and Weissauer explain in [KW] III.15, in our situation the conditions i) and ii) in the definition are automatically satisfied; more precisely, given any isomorphism a^{∗}F ⟶ p_{2}^{∗}F, one can change it into a rigidified isomorphism, and every rigidified isomorphism satisfies the cocycle condition. (Here we use that the Iwahori acts through geometrically connected quotients on suitable finite pieces of the affine flag variety.)

Similarly, on the affine Grassmannian we have an action of the maximal parahoric subgroup K = G(O), where O = ℤ_{p} in the unequal characteristic case, and O = 𝔽_{p}[[t]] in the function field case.

Proposition 2.3 If F is a K-equivariant perverse sheaf on Grass, then its sheaf of nearby cycles RΨ(F) is B-equivariant.

Proof. In the function field case, this is [Ga] Prop. 4; for the unequal characteristic case see [HN1] §4. □

We denote by P^{B}(Fl, \overline{ℚ}_{ℓ}) the category of B-equivariant perverse sheaves on Fl/𝔽_{p}. Later on (section 4) we will consider B-equivariant perverse sheaves which have Galois structure compatible with the group action. We postpone the discussion of those sheaves to that section.

We note that the intersection complexes IC_{μ} on Grass are G(O)-equivariant, and the intersection complexes IC_{w} on Fl are B-equivariant.

2.7 The Kottwitz conjecture

For any F ∈ D_{c}^{b}(Fl ×_{s} η, \overline{ℚ}_{ℓ}), one can define its semi-simple trace function

x ↦ Tr^{ss}(Fr_{q}, F_{x}),

for any x ∈ Fl(𝔽_{q}); cf. [HN1]. Here Fr_{q} ∈ Γ denotes an arbitrary geometric Frobenius (the semi-simple trace is independent of the choice). If F is a suitably B-equivariant object (e.g., an object of P^{B}(Fl ×_{s} η, \overline{ℚ}_{ℓ}), cf. section 4.1), then this function defines an element in the Iwahori-Hecke algebra

H = C_{c}(B\G(𝔽_{q}((t)))/B)

of B-bi-invariant compactly-supported \overline{ℚ}_{ℓ}-valued functions on G(𝔽_{q}((t))). (Remark: in the sequel, we will often conflate the notions of Iwahori-Hecke algebras and affine Hecke algebras without lingering on the difference.)

In the p-adic setting, fix a dominant coweight μ for the group G = GL_{n} or GSp_{2n}, and the associated model M_{μ} from [HN1]. For the nearby cycles sheaf RΨ := RΨ^{M_{μ}}(IC_{μ}), the Kottwitz conjecture is the following equality of functions in the Iwahori-Hecke algebra for G:

Tr^{ss}(Fr_{q}, RΨ) = ε_{μ}q_{μ}^{1/2} ∑_{λ≤μ} m_{μ}(λ)z_{λ}.

Here, ε_{μ} = (−1)^{ℓ(μ)}, q_{μ} = q^{ℓ(μ)}, λ ranges over dominant coweights preceding μ in the usual partial order, m_{μ}(λ) is the multiplicity of λ in the character of the dual group having highest weight μ, and z_{λ} is the Bernstein function attached to λ. More precisely z_{λ} = ∑_{λ∈W μ} Θ_{λ}, where Θ_{λ} is the function T̃_{t_{λ_{1}}} T̃_{t_{λ_{2}}}^{−1}; here

λ = λ_{1} − λ_{2} and λ_{i} is dominant, and T̃_{x} := q_{x}^{−1/2}T_{x} for x ∈ W̃. The symbol T_{x} denotes the generator char(BxB) in the Iwahori-Hecke algebra.

This formula was proved in [HN1] in the situation at hand. In the function field case, the analogous formula holds. In that analogue, IC_{μ} is replaced with the exterior product IC_{μ} ⊠ δ on Grass × G/B, where δ is the skyscraper sheaf supported at the base point of G/B. This is a consequence of Gaitsgory’s work [Ga], and holds for any connected reductive group G.

We make extensive use of this formula, in particular in the computation of the examples in section 9, and in the proof that RΨ satisfies the property (P) (cf. section 6).

3 Inverse Kazhdan-Lusztig polynomials

3.1 The sheaf IC(B\overline{v})

Let us recall some of the notation introduced above. Let k = 𝔽̄_{p}. We denote by B the Iwahori subgroup of G(k((t))), i.e. the inverse image of the fixed Borel group B under the projection G(k[[t]]) ⟶ G. By N^{−} we denote the inverse image of the unipotent radical of the opposite Borel under the projection G(k[t^{−1}]) ⟶ G. Furthermore, let N^{−}(n) be the inverse image of T (k[t^{−1}]/t^{−n}) under the projection N^{−} ⟶ G(k[t^{−1}]/t^{−n}) (cf. [F]). Note that Lie(N^{−}) is generated by the negative affine roots (recall we embed X_{∗}(T) ↪ T (k((t))) by λ ↦ λ(t)).

For elements w and y in the extended affine Weyl group W̃, we denote by B_{y} the B-orbit of y, by B^{w} the N^{−}-orbit of w, by B\overline{w} its closure, by B_{y}^{w} the intersection B_{y} ∩ B^{w}, and by B\overline{w} ; y the intersection B_{y} ∩ B\overline{w}. Recall the basic fact (cf. [F]) that B_{y}^{w} ≠ ∅ ⇔ w ≤ y in the Bruhat order on W̃.

Further, we define the opposite Iwahori subgroup B^{−} to be the inverse image of the opposite Borel subgroup B^{−} under the projection G(k[t^{−1}]) → G(k). Note that N^{−} ◁ B^{−} and B^{−} = N^{−} · (B^{−} ∩ T (k((t)))). Hence for any w ∈ W̃, N^{−}w∗ = B^{−}w∗, where ∗ denotes the base point in Fl.

Fix w ∈ W̃ such that Adm(μ) ⊆ {v | v ≤ w}.

Let Ω = ⋃_{v≤w} vN^{−}∗. This is an open subset of Fl.

Lemma 3.1 We have Ω = ∐_{v≤w} B^{v}. Thus for y ≤ w we have B_{y} ⊆ Ω.

Proof. We define

N_{w}^{−} = N^{−} ∩ wN^{−}w^{−1}
N_{w}^{+} = N^{+} ∩ wN^{−}w^{−1}.

Here N^{+} denotes the preimage of N under the projection G(k[[t]]) → G(k); Lie(N^{+}) is generated by the positive affine roots.

We then have

N^{−} ≅ N_{w}^{−} × ∏_{{a<0 \atop w^{−1}(α)>0}} U_{α},

and

N_{w}^{+} = ∏_{{α>0 \atop w^{−1}(α)<0}} U_{α} = w ( ∏_{{α<0 \atop w(α)>0}} U_{α}) w^{−1}.

Furthermore, it is easy to see that B_{w} = N_{w}^{+}w∗ and B^{w} = N_{w}^{−}w∗.

From the above we have

vN^{−} = vN_{v^{−1}}^{−} · (v^{−1}N_{v}^{+}v) = N_{v}^{−}N_{v}^{+}v.

Thus clearly

B^{v} = N_{v}^{−}v∗ ⊆ vN^{−}∗,

so

∐_{v≤w} B^{v} ⊆ ⋃_{v≤w} vN^{−} ∗ .

On the other hand, N_{v}^{+}v = B_{v} ⊆ ∐_{v′≤v} B^{v′}, hence N_{v}^{−}N_{v}^{+}v ⊆ ∐_{v′≤v} B^{v′} and thus

⋃_{v≤w} vN^{−}∗ ⊆ ∐_{v≤w} B^{v}.

□

For n sufficiently large, the quotient N^{−}(n)\Ω exists; cf. [F] (choose n large enough so that N^{−}(n) ⊂ N_{v}^{−}, for every v ≤ w). The reason is that we have a product decomposition

vN^{−}∗ = N_{v}^{−}N_{v}^{+}v∗ ≅ N^{−}(n) × 𝔸^{a} × 𝔸^{b},

where b = dim(N_{v}^{+}) = ℓ(v), and a is the number of affine roots in N_{v}^{−} which are not in N^{−}(n); cf. [F]. Thus we see that more precisely, the quotient N^{−}(n)\Ω can be covered by affine spaces, and in particular is smooth.

In the following discussion, we often tacitly identify B_{y} = N_{y}^{+}y, B^{y} = N_{y}^{−}y and B^{y} × B_{y} = N_{y}^{−}N_{y}^{+}y = yN^{−}∗. We have a commutative diagram (v ≤ y ≤ w)

\xymatrix{B\overline{v} ; y \ar[d]^{≅} \ar@{^{(}->}[r] & B_{y} \ar[d]^{≅} \ar@{^{(}->}[r] & B^{y} × B_{y} \ar[d] \ar@{^{(}->}[r] & Ω \ar[d]^{π} \\ π(B\overline{v} ; y) \ar@{^{(}->}[r] & π(B_{y}) \ar@{^{(}->}[r] & N^{−}(n)\B^{y} × B_{y} \ar@{^{(}->}[r] & N^{−}(n)\Ω.}

Note that

π(B\overline{v} ; y ∩ Ω) = π(B_{y}) ∩ π(B\overline{v} ∩ Ω).

3.1 The sheaf IC(B\overline{v}) 13

Furthermore π(B^{v} ∩ Ω) is isomorphic to an affine space (thus in particular is smooth), and its codimension in N^{−}(n)\Ω is ℓ(v). Finally, we have

N^{−}(n)\(B^{y} × B_{y}) = (N^{−}(n)\B^{y}) × B_{y} =: A_{y}

and this is an open subset of N^{−}(n)\Ω, and A_{y} ∩π(B\overline{v} ∩Ω) = (N^{−}(n)\B^{y})×B\overline{v} ; y is an open subset of π(B\overline{v} ∩ Ω).

By taking the quotient of the morphism N^{−} × Ω ⟶ Ω × Ω, (g, x) ↦ (x, gx) giving the action of N^{−} on Ω, we get the groupoid

N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω ⟶ N^{−}(n)\Ω × N^{−}(n)\Ω

(cf. [F]).

Proposition 3.2 This is a smooth groupoid, i.e. the maps N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω ⟶ N^{−}(n)\Ω, (g, x) ↦ x, and N^{−}(n)\N^{−}×^{N^{−}(n)}Ω ⟶ N^{−}(n)\Ω, (g, x) ↦ gx, are smooth.

Proof. Consider the maps

ϕ_{1}, ϕ_{2} : N^{−} × Ω ⟶ Ω, (g, x) ↦ x, resp. (g, x) ↦ gx.

These are formally smooth maps since N^{−} is formally smooth. They induce maps

ψ_{1}, ψ_{2} : N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω ⟶ N^{−}(n)\Ω,

where ψ_{1}(g, x) = x, ψ_{2}(g, x) = gx.

We must show that ψ_{1} and ψ_{2} are smooth. We have a commutative diagram

\xymatrix{N^{−} × Ω \ar[r]^{ϕ,fs} \ar[d]^{fs} & Ω \ar[d]^{fs} \\ N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω \ar[r]^{ψ} & N^{−}(n)\Ω}

where ϕ is either ϕ_{1} or ϕ_{2}, ψ is the corresponding map and ’fs’ means that we already know that this morphism is formally smooth.

Since N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω ⟶ N^{−}(n)\Ω is a morphism of finite type between Noetherian schemes (over \overline{𝔽}_{p}), in order to prove that it is smooth, it is sufficient to show the infinitesimal lifting criterion for local Artin rings with residue class field \overline{𝔽}_{p}. (see [SGA 1], Exp. III Thm. 3.1).

So let A be a local Artin ring with residue field \overline{𝔽}_{p}, let I ⊂ A be a nilpotent ideal, and look at a diagram

\xymatrix{Spec A/I \ar[r] \ar[d] & N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω \ar[d] \\ Spec A \ar[r] & N^{−}(n)\Ω.}

We have to find a morphism Spec A ⟶ N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω which makes the diagram commutative.

Since the morphism N^{−}×Ω ⟶ N^{−}(n)\N^{−}×^{N^{−}(n)}Ω is surjective on \overline{𝔽}_{p}-valued points, we can extend the previous diagram to a diagram

\xymatrix{Spec \overline{𝔽}_{p} \ar[r] \ar[d] & N^{−} × Ω \ar[d]^{ξ} \\ Spec A/I \ar[r] \ar[d] & N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω \ar[d]^{ψ} \\ Spec A \ar[r] & N^{−}(n)\Ω.}

Now since ξ is formally smooth, we can insert a map Spec A/I ⟶ N^{−} ×Ω into the diagram. Then we get a map Spec A ⟶ N^{−} × Ω, since the composition ψξ is formally smooth. Now we can define the map we are looking for as the composition of this map with ξ.

Thus we see that ψ_{1} and ψ_{2} are indeed smooth. □

From the Proposition, we immediately get the following corollary which will be used in the proof of the theorem of Kazhdan and Lusztig.

Corollary 3.3 Let v ≤ z ≤ w. For all z′ ∈ B^{z}(𝔽_{q}),

IC_{z′}(π(B\overline{v} ∩ Ω)) ≅ IC_{z}(π(B\overline{v} ∩ Ω)).

Proof. By pull-back the maps ψ_{1} and ψ_{2} in the Proposition induce smooth maps

ψ_{1}, ψ_{2} : N^{−}(n)\N^{−} ×^{N^{−}(n)} (B\overline{v} ∩ Ω) ⟶ π(B\overline{v} ∩ Ω),

where ψ_{1}(g, x) = x, ψ_{2}(g, x) = gx.

Now choose g ∈ N^{−} such that gz = z′. The point (g, z) maps to z resp. z′ under ψ_{1} resp. ψ_{2}. Since both these maps are smooth, the corollary follows. □

However, we can even say more: The sheaf IC(π(B\overline{v} ∩ Ω)) on π(B\overline{v} ∩ Ω) ⊆ N^{−}(n)\Ω is equivariant under the above groupoid (our definition of equivariance obviously makes sense in this context). Namely, we have

ψ_{1}^{∗}IC(π(B\overline{v} ∩ Ω)) ≅ IC(N^{−}(n)\N^{−} ×^{N^{−}(n)} (B\overline{v} ∩ Ω)) ≅ ψ_{2}^{∗}IC(π(B\overline{v} ∩ Ω)),

since ψ_{1} and ψ_{2} are both smooth. We can choose an isomorphism which is rigidified along the zero section, and since N^{−}(n)\N^{−} is connected such an isomorphism will automatically satisfy the cocycle condition.

Thus the pull-back of the sheaf IC(π(B\overline{v} ∩ Ω)) to Ω is equivariant with respect to the action of N^{−}.

We can now define the intersection complex IC(B\overline{v}) of the anti-cell B\overline{v}. Since B\overline{v} is just an ind-scheme, it is not a priori clear how to make sense of that. However, by looking at the quotients N^{−}(n)\Ω, we can define a sheaf on Fl which merits the name IC(B\overline{v}).

The pull-back of the sheaf IC(π(B\overline{v} ∩ Ω)) to Ω is independent of n: in fact, let n′ > n, and consider the diagram

\xymatrix{Ω \ar[rr]^{π} \ar[dr]^{π′} &  & N^{−}(n)\Ω \ar[dl]_{ψ} \\  & N^{−}(n′)\Ω.}

The map ψ is an 𝔸^{N}-bundle (for suitable N), and π(B\overline{v} ∩Ω) = ψ^{−1}(π′(B\overline{v} ∩Ω)). Thus the pull-back of IC(π′(B\overline{v} ∩Ω)) along ψ is just IC(π(B\overline{v} ∩Ω)). This means that the pull-back of IC(π(B\overline{v}∩Ω)) to Ω is independent of n. Since it is obviously compatible with enlarging Ω, we indeed get a sheaf on Fl, which we denote by IC(B\overline{v}).

We will only ever need to work with the restriction of IC(B\overline{v}) to finitedimensional subschemes of Fl. In particular we do not really need to define this sheaf on Fl — we could always work on a suitable quotient N^{−}(n)\Ω.

We will need the following property of IC(B\overline{v}). When restricted to an intersection B\overline{v} ; y, we get the intersection complex of that space:

IC(B\overline{v})|_{B\overline{v} ; y} = IC(B\overline{v} ; y).

This follows from

Proposition 3.4 We have

IC(π(B\overline{v} ∩ Ω))|_{π(B\overline{v} ; y)} = IC(B\overline{v} ; y).

Proof. This follows immediately from the fact that A_{y} ∩ π(B\overline{v} ∩ Ω) = (N^{−}(n)\B^{y}) × B\overline{v} ; y is an open subset of π(B\overline{v} ∩ Ω) and that N^{−}(n)\B^{y} is just an affine space. □

3.2 The theorem of Kazhdan and Lusztig

We start with two lemmas, and then prove the theorem of Kazhdan and Lusztig.

Lemma 3.5 We have

#B_{y}^{z}(𝔽_{q}) = R_{z,y}(q).

Proof. See [Had], section 2. □

Lemma 3.6 On A_{y}, which is isomorphic to an affine space 𝔸^{N}, we have a 𝔾_{m}-action given by

λ · (z_{1}, …, z_{N}) = (λ^{a_{1}}z_{1}, …, λ^{a_{N}} z_{N}),

for certain a_{i} > 0, which preserves π(B\overline{v} ∩ Ω) ∩ A_{y} and π(B\overline{v} ∩ B_{y}).

Proof. We decompose A_{y} = N^{−}(n)\B^{y} × B_{y}, and we will define 𝔾_{m}-actions on the two factors separately. With respect to this decomposition, we have π(B\overline{v} ∩ Ω) ∩ A_{y} = N^{−}(n)\B^{y} × B\overline{v} ∩ B_{y} and π(B\overline{v} ∩ B_{y}) = \{y\} × B\overline{v} ∩ B_{y}. Thus the 𝔾_{m}-action on the first factor must just be a contracting 𝔾_{m}-action which fixes the origin.

To define the action on the second factor, consider the action of the torus T (k) on B_{y}. Obviously it fixes B\overline{v} ∩ B_{y}, so we only have to find a cocharacter ϕ : 𝔾_{m} ⟶ T such that the induced 𝔾_{m}-action is contracting.

This is possible because the set of finite roots α such that for some n, the affine root (α, n) ’occurs’ in B_{y} is the union of the sets

{α > 0 | ⟨α, λ⟩ ≤ 0, and ỹα < 0 if ⟨α, λ⟩ = 0}

and

{α < 0 | ⟨α, λ⟩ ≤ −1, and ỹα < 0 if ⟨α, λ⟩ = −1},

where y^{−1} = ỹt_{λ}, ỹ ∈ W, λ ∈ X_{∗}(T). We may put ϕ = −mλ − ỹ^{−1}ρ^{∨} for any sufficiently large integer m. □

Theorem 3.7 (Interpretation of inverse Kazhdan-Lusztig polynomials. ([KL2], Prop. 5.7))

Let v ≤ y. Then IH^{i}(B\overline{v} ; y) = 0 for i odd, and

Q_{v,y}(q) = ∑_{i≥0} dim IH^{2i}(B\overline{v} ; y)q^{i} = Tr(Fr_{q}, IH^{•}(B\overline{v} ; y)).

Remark. This result is stated in [KL2] as a proposition which ’should not be difficult to prove’.

Kashiwara and Tanisaki give a proof of the analog of this result in the setting of mixed Hodge modules in [KT], Thm. 6.6.4; in particular they work over the complex numbers.

Proof. Given the lemmas above, the theorem is proved much in the same way as Theorem 4.2 in [KL2] which gives an analogous interpretation for the Kazhdan-Lusztig polynomials in the finite dimensional case.

Choose Ω = ⋃_{v≤w} vN^{−}∗ sufficiently large, and denote the projection Ω ⟶ N^{−}(n)\Ω by π as before.

We write B_{?}^{?} as an abbreviation for π(B_{?}^{?} ∩ Ω). Note that B_{y}^{v} ≅ B_{y}^{v}. Denote by ℋ ^{i}(B\overline{v}) the i-th cohomology sheaf of IC(B\overline{v}), and by ℋ_{z}^{i}(B\overline{v}) its stalk in a point z.

First of all, Lemma 4.5a) in [KL2] together with Lemma 3.6, the fact that B^{y} × B\overline{v} ; y is open in B\overline{v}, and the fact that B^{y} is just an affine space, yield that IH^{i}(B\overline{v} ; y) ≅ ℋ_{y}^{i}(B\overline{v}). In the sequel we will thus always work with ℋ_{y}^{i}(B\overline{v}).

For z ≥ v consider the property

P(z): For any z′ ∈ B^{z}(𝔽_{q}), we have ℋ_{z′}^{i}(B\overline{v}) = 0 for odd i, and for even i all eigenvalues of Fr_{q} on ℋ_{z′}^{i}(B\overline{v}) are equal to q^{i/2}.

Obviously P(v) is true. Now fix y > v and assume that P(z) holds for all v ≤ z < y. We want to show that this implies P(y).

By assumption (B\overline{v} ∩ A_{y}) − B^{y} is very pure. Since

(B\overline{v} ∩ A_{y}) − B^{y} ≅ B^{y} × (B\overline{v} ; y − \{y\})

and B^{y} is smooth, we see that B\overline{v} ; y − \{y\} is very pure. Thus by Lemma 3.6 and Lemma 4.5b) in [KL2], B\overline{v} ; y is very pure. This implies that B\overline{v} ∩ A_{y} is very pure.

Now the Lefschetz trace formula implies (use Lemma 3.5 and Lemma 3.3)

Tr(Fr_{q}, IH_{c}^{•}(B\overline{v} ∩ A_{y}))
= ∑_{v≤z≤y} ∑_{z′∈(B^{z}∩A_{y})(𝔽_{q})}Tr(Fr_{q}, ℋ_{z′}^{•}(B\overline{v}))
= ∑_{v≤z≤y} q^{dim B^{y}}R_{z,y}(q)Tr(Fr_{q}, ℋ_{z}^{•}(B\overline{v})).

By Poincaré duality we have

Tr(Fr_{q}, IH_{c}^{•}(B\overline{v} ∩ A_{y})) = q^{dim(B\overline{v}∩A_{y})}Tr(Fr_{q}^{−1}, IH^{•}(B\overline{v} ∩ A_{y})),

and [KL2], Lemma 4.5 gives us

Tr(Fr_{q}^{−1}, IH^{•}(B\overline{v} ∩ A_{y})) = Tr(Fr_{q}^{−1}, ℋ_{y}^{•}(B\overline{v})).

Altogether, we now get

q^{ℓ(y)−ℓ(v)}Tr(Fr_{q}^{−1}, ℋ_{y}^{•}(B\overline{v})) = ∑_{v≤z≤y} R_{z,y}Tr(Fr_{q}, ℋ_{z}^{•}(B\overline{v})).  (1)

This equation is completely analogous to (4.6.4) in [KL2], and exactly the same arguments as in loc. cit. now yield that, ∀i,

Tr(Fr_{q}, ℋ_{y}^{i}(B\overline{v})) ∈ ℤ[q],

which implies that P(y) holds. This proves the second equality in the statement of the theorem.

Furthermore, since the inverse Kazhdan-Lusztig polynomials are characterized by

Q_{v,v} = 1, q^{ℓ(y)−ℓ(v)}Q_{v,y}(q^{−1}) = ∑_{v≤z≤y} R_{z,y}Q_{v,z},

deg Q_{v,y} ≤ (l(y) − l(v) − 1)/2, (v < y)

we also get the first equality in the statement of the theorem from equation (1). □

4 The main theorem

4.1 Decomposition of B-equivariant sheaves

We begin with a discussion of the categories of perverse sheaves to which our results apply. We will denote by P (Fl×_{s} η, \overline{ℚ}_{ℓ}) the full subcategory of D_{c}^{b}(Fl×_{s} η, \overline{ℚ}_{ℓ}) consisting of middle perverse sheaves endowed with a compatible action of the Galois group Γ = Gal(η̄/η). (For the precise meaning of “compatible”, see section 5.1). We let P^{B}(Fl×_{s}η, \overline{ℚ}_{ℓ}) denote the full subcategory of B-equivariant objects of P (Fl×_{s}η, \overline{ℚ}_{ℓ}): these are perverse sheaves on Fls̄ which are equivariant for Γ and B, in a compatible way.

Similarly, we let P_{Weil}(Fl, \overline{ℚ}_{ℓ}) denote the category of pairs (F, F_{q}^{∗}) where F is a perverse sheaf on Fls̄ and F_{q}^{∗} : Fr_{q}^{∗}F ˜→F is an isomorphism of perverse sheaves (a Weil structure on F). Further, let P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) denote the full subcategory of B-equivariant objects of P_{Weil}(Fl, \overline{ℚ}_{ℓ}): the B-equivariance of F is assumed compatible with F_{q}^{∗}.

Lemma 4.1 The subcategory P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) of P (Fl, \overline{ℚ}_{ℓ}) is stable under formation of kernels and cokernels. Objects in this subcategory have finite length. The irreducible objects are intersection complexes of the form IC(L)[ℓ(w)] = j_{!∗}L[ℓ(w)], where j : B_{w} → Fl is the locally closed immersion of an Iwahoriorbit B_{w} in Fl, and L is a B-equivariant (thus geometrically constant) irreducible Weil sheaf on B_{w}.

Proof. The statements concerning kernels and cokernels, and finite length, are clear. We prove that all the irreducible subquotients of F ∈ P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) taken in this category are of the required form IC(L)[ℓ(w)], by Noetherian induction on supp(F). Indeed, we may find an open immersion j : B_{w} → supp(F) such that j^{∗}F is a (necessarily B-equivariant) Weil-sheaf on B_{w} (up to a shift by ℓ(w)). Now the kernel and cokernel of the adjunction map j_{!}j^{∗}F → F belong to P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) and are supported on proper closed B-invariant subsets of supp(F). Further, the kernel of the canonical surjection j_{!}j^{∗}F → j_{!∗}j^{∗}F also has this property. Since for the action of B on the orbit B_{w}, the stabilizer of any point is geometrically connected, any B-equivariant Weil sheaf on B_{w} is geometrically just a constant sheaf.

The fact that the subquotients of F have the required form is a consequence of these remarks. (Note that j_{!} etc. denotes the derived functor here. We have used the fact that the morphism j is affine to ensure that j_{!} preserves perversity and that j_{!∗} is a quotient of j_{!}.)

It also follows from this that the irreducible objects of P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) are of the stated form. □

Corollary 4.2 Objects in P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) are mixed.

Now consider a Weil-perverse sheaf F ∈ P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) such that for all x ∈ Fl(𝔽_{q}), we have Tr(Fr_{q}, F_{x}) ∈ ℤ[q^{1/2}, q^{−1/2}]. We will denote the full subcategory of those sheaves by P_{√\overline{q}}^{B}(Fl, \overline{ℚ}_{ℓ}). Similarly, let P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) denote the full subcategory of sheaves such that Tr(Fr_{q}, F_{x}) ∈ ℤ[q, q^{−1}] for all x ∈ Fl(𝔽_{q}). The category P_{√\overline{q}}^{B}(Fl, \overline{ℚ}_{ℓ}) gives rise via the sheaf-function dictionary precisely to elements in the Iwahori-Hecke algebra (for this reason it is sometimes called the Hecke category).

Lemma 4.3 The subcategory P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) in P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) is stable under extensions, quotients, and subquotients, and under Verdier duality. The irreducible objects of P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) are of the form IC_{w}(i), for w ∈ W̃, i ∈ ℤ.

Proof. The statements concerning extensions, quotients, and Verdier duality are clear. Let F ∈ P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}). We prove all of its irreducible subquotients in the category P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) are of the form IC_{w}(i), by Noetherian induction on supp(F). Replacing F by its semi-simplification, we may assume F is semisimple. Choose an open immersion j : B_{w} → supp(F), such that j^{∗}F[−ℓ(w)] is a (geometrically constant) Weil sheaf L. By hypothesis, Tr(Fr_{q}, L_{w}) ∈ ℤ[q, q^{−1}]. The linear independence of characters shows that the eigenvalues of Frobenius on L_{w} are of form q^{i}, for various integers i, in other words, L is a sum of Weil sheaves on B_{w} of the form \overline{ℚ}_{ℓ}(i), i ∈ ℤ. By the previous Lemma, the intersection complex IC(L)[ℓ(w)] is a subquotient of F in the category P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}). The quotient F/IC(L[ℓ(w)]) belongs to P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) (because by Kazhdan-Lusztig [KL2], IC(L[ℓ(w)]) ∈ P_{q}^{B}(Fl, \overline{ℚ}_{ℓ})), and has support a proper closed subset of supp(F). Applying the induction hypothesis to this quotient proves that all the irreducible subquotients of F have the required form.

The fact that the subcategory P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) of P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) is closed under formation of subobjects follows. □

Corollary 4.4 Let F be an object of P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}). Then the constituents of the Jordan-Hölder series of F have the form IC_{w}(−i), w ∈ W̃, i ∈ ℤ. Each of them appears with a certain multiplicity, which we denote by m(F, w, i).

4.2 Relation to the Hecke algebra

Denote by H = ⨁_{w∈W̃} ℤ[q^{−1/2}, q^{1/2}]T_{w} the Iwahori-Hecke algebra associated to G.

As usual, we write q_{w} = q^{ℓ(w)}, ε_{w} = (−1)^{ℓ(w)}.

A perverse sheaf F ∈ P_{√\overline{q}}^{B}(Fl, \overline{ℚ}_{ℓ}) gives rise to a function Tr(Fr_{q}, F) = ∑_{w} Tr(Fr_{q}, F_{w})T_{w} ∈ H, and we have a sheaf-function dictionary à la Grothendieck.

On the other hand, Kazhdan and Lusztig have defined a different basis C_{w} of H. We slightly change the normalization, and work with elements C′′_{w} instead.

They are related to the C_{w} resp. to the elements C′_{w} sometimes appearing in the literature as follows. By definition, C′′_{w} is the function Tr(Fr_{q}, IC(B_{w})[ℓ(w)]). Also by definition, C′_{w} = Tr(Fr_{q}, IC(B_{w})(ℓ(w)/2)), so we have C′′_{w} = ε_{w}q_{w}^{1/2}C′_{w}. But we also have σ(C_{w}) = ε_{w}C′_{w}, where σ is the involution of the Hecke algebra given by q ↦ q^{−1}, T_{w} ↦ ε_{w}q_{w}^{−1}T_{w}. Thus C′′_{w} = q_{w}^{1/2}σ(C_{w}). The base change matrix giving the relation between the T_{w} and the C′′_{w} is given by Kazhdan-Lusztig polynomials resp. inverse Kazhdan-Lusztig polynomials:

T_{w} = ∑_{x} ε_{w}Q_{x,w}C′′_{x},
C′′_{w} = ∑_{x} ε_{w}P_{x,w}T_{x}.

Now if F ∈ P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}), the multiplicities m(F, w, i) defined above are just the coefficients of Tr(Fr_{q}, F) with respect to the basis C′′_{w}:

Tr(Fr_{q}, F) = ∑_{w} (∑_{i} m(F, w, i)q^{i}) C′′_{w}.

Thus, knowing the function for F explicitly is equivalent to knowing all the multiplicities m(F, w, i) for F.

For later use note that the identity \overline{C′′_{w}} = q_{w}^{−1}C′′_{w}, together with the fact that Verdier duality descends to the Kazhdan-Lusztig involution on the Iwahori-Hecke algebra, gives the following identity.

Lemma 4.5 We write m(F, w) = ∑_{i} m(F, w, i)q^{i} for F ∈ P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}). Then

m(DF, w) = q_{w}^{−1}\overline{m(F, w)}.

4.3 The property (P)

We say that a sheaf F ∈ P_{√\overline{q}}^{B}(Fl, \overline{ℚ}_{ℓ}) satisfies the property (P) (for “palindromic”), if for all y ∈ Fl(𝔽_{q}) we have

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

for some d ∈ ℤ (setting ε_{d} = (−1)^{d}). Here DF denotes the Verdier dual of F, and h ↦ h̄ is the involution of ℤ[q^{1/2}, q^{−1/2}] determined by q^{1/2} ↦ q^{−1/2}.

In other words, we have

Tr(Fr_{q}, DF) = ∑_{y} ε_{d}ε_{y}q^{−d}q_{y}^{−1}\overline{Tr(Fr_{q}, DF_{y})}T_{y}

in the Iwahori-Hecke algebra. By applying Verdier duality (i.e. by applying the Kazhdan-Lusztig involution \overline{·}) and using the definition \overline{T}_{y} = T_{y^{−1}}^{−1}, we get

Tr(Fr, F) = ∑_{x} (∑_{y≥x} ε_{d}q^{d}Tr(Fr_{q}, DF_{y})ε_{x}R_{x,y}(q)) T_{x}.

Therefore we see that property (P) implies

Tr(Fr_{q}, F_{x}) = ε_{d}q^{d} ∑_{y≥x} ε_{x}Tr(Fr_{q}, DF_{y})R_{x,y}(q).

Remark. We will see in section 6 that RΨ satisfies (P). More generally, the notion of Wakimoto sheaf allows us to define a much larger class of sheaves that satisfy (P); cf. section 7.

Arguing as above, it is easy to see from the definitions that property (P) is stable under Verdier duality.

Lemma 4.6 A sheaf F ∈ P_{√\overline{q}}^{B}(Fl, \overline{ℚ}_{ℓ}) satisfies (P) for an integer d if and only if DF satisfies (P) for −d.

Proof. Use the identities

Tr(Fr_{q}, F_{x}) = ∑_{x≤y} ε_{x}ε_{y}q_{y}^{−1}R_{x,y}\overline{Tr(Fr_{q}, DF_{y})}
\overline{Tr(Fr_{q}, F_{x})} = ∑_{x≤y} q_{x}R_{x,y}Tr(Fr_{q}, DF_{y}).

□

4.4 Cohomological interpretation of the multiplicities

In this section we will give a cohomological interpretation of the multiplicities associated to a sheaf F which satisfies the property (P). This can be seen as a generalization of the theorem of Kazhdan and Lusztig which gives an interpretation of the inverse Kazhdan-Lusztig polynomials (Theorem 3.7). Their theorem is the key ingredient of the proof.

Theorem 4.7 Suppose that F ∈ P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}) satisfies (P). Then

∑_{i} m(F, w, i)q^{i} = ε_{d}q^{d}Tr(Fr_{q}, H_{c}^{•}(Fl, DF ⊗ IC(B\overline{w})))
= ε_{d}q^{d−ℓ(w)}Tr(Fr_{q}, H_{c}^{•}(Fl, F ⊗ IC(B\overline{w}))^{∨}).

Here, V^{∨} = Hom(V, \overline{ℚ}_{ℓ}) for a \overline{ℚ}_{ℓ}-space V.

Proof. Using the definition of the multiplicities and the fact that F satisfies (P), we have

∑_{i} m(F, w, i)q^{i} = ∑_{w≤x} ε_{x}Tr(Fr_{q}, F_{x})Q_{w,x}
= ∑_{w≤x} ∑_{x≤y} ε_{d}q^{d}Tr(Fr_{q}, (DF)_{y})Q_{w,x}R_{x,y}.

Rearranging this and using the formula ∑_{w≤x≤y} Q_{w,x}R_{x,y} = q_{y}q_{w}^{−1}Q_{w,y}(q^{−1}), we get

∑_{i} m(F, w, i)q^{i} = ε_{d}q^{d} ∑_{w≤y} Tr(Fr_{q}, (DF)_{y}) ∑_{w≤x≤y} Q_{w,x}R_{x,y}
= ε_{d}q^{d} ∑_{w≤y}(q_{y}q_{w}^{−1}Q_{w,y}(q^{−1}))Tr(Fr_{q}, (DF)_{y}).

By using the Poincaré dual of Theorem 3.7, we see that

q_{y}q_{w}^{−1}Q_{w,y}(q^{−1}) = Tr(Fr_{q}, IH_{c}^{•}(B_{y} ∩ B\overline{w})).

Thus we get

∑_{i} m(F, w, i)q^{i}

= ε_{d}q^{d} ∑_{w≤y} Tr(Fr_{q}, IH_{c}^{•}(B_{y} ∩ B\overline{w}))Tr(Fr_{q}, (DF)_{y})
= ε_{d}q^{d} ∑_{w≤y} Tr(Fr_{q}, H_{c}^{•}(B_{y} ∩ B\overline{w}, IC(B_{y} ∩ B\overline{w}) ⊗ DF|_{B_{y}∩B\overline{w}}))
= ε_{d}q^{d}Tr(Fr_{q}, H_{c}^{•}(Fl, IC(B\overline{w}) ⊗ DF))),

taking Proposition 3.4 into account. This proves the first equality of the theorem.

The second equality can be proved by applying the first equality to the sheaf DF in place of F (cf. Lemma 4.6), and by using Lemma 4.5. □

Remark. It is clear from the proof of the theorem that we could replace the condition that F satisfies (P) by the following: Let F̃ be a B-equivariant sheaf such that for all y

Tr(Fr_{q}, DF_{y}) = ε_{y}q_{y}^{−1}\overline{Tr(Fr_{q}, F̃_{y})}.

Then

∑_{i} m(F, w, i)q^{i} = Tr(Fr_{q}, H_{c}^{•}(Fl, F̃ ⊗ IC(B\overline{w}))).

4.5 The Kazhdan-Lusztig theorem as a consequence of the main theorem

Fix y ∈ W̃, and consider the sheaf F = j_{y!}\overline{ℚ}_{ℓ}[ℓ(y)]. This belongs to P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}), and satisfies property (P) with d = ℓ(y). We may therefore apply the second equality of Theorem 4.7 to this sheaf. Using the projection formula for j_{y!}, we have

ε_{y}q_{y}q_{w}^{−1}Tr(Fr_{q}, H_{c}^{•}(Fl, F ⊗ IC(B\overline{w}))^{∨}) = q_{y}q_{w}^{−1}Tr(Fr_{q}, H_{c}^{•}(Fl, j_{y!}(j_{y}^{∗}IC(B\overline{w})))^{∨})
= q_{y}q_{w}^{−1}Tr(Fr_{q}, H_{c}^{•}(Bw̄ ; y, IC(B\overline{w} ; y))^{∨})
= q_{y}q_{w}^{−1}Tr(Fr_{q}^{−1}, H_{c}^{•}(Bw̄ ; y, IC(B\overline{w} ; y)))
= Tr(Fr_{q}, IH^{•}(B\overline{w} ; y)),

where the last equality follows from Poincaré duality. Since ∑ m(F, w, i)q^{i} = Q_{w,y}(q), this is just one version of the theorem of Kazhdan and Lusztig (Theorem 3.7). (Taking into account the very-purity of B\overline{w} ; y, one can recover the full statement of that theorem.)

4.6 Consequences

4.6.1 w ∈ (W̃/W)_{min}

Denote by π : Fl ⟶ Grass the projection. The map π is smooth and proper as a morphism of ind-schemes. We denote the projection W̃ ⟶ W̃/W by π, too.

If w ∈ (W̃/W)_{min}, we have B\overline{w} = π^{−1}(Q\overline{π(w)}), and so IC(Bw̄) = π^{∗}IC(Q\overline{π(w)}). Here Q\overline{π(w)} is the closure of the N^{−}-orbit of π(w) in the affine Grassmannian for G.

Thus in this case we can simplify the formula in the theorem by considering the push-forward to Grass and applying the projection formula. We get

∑_{i} m(F, w, i)q^{i} = ε_{d}q^{d}Tr(Fr_{q}, H_{c}^{•}(Fl, DF ⊗ IC(B\overline{w})))
= ε_{d}q^{d}Tr(Fr_{q}, H_{c}^{•}(Grass, (π_{∗}DF) ⊗ IC(Q\overline{π(w)}))).

We remark that this formula simplifies further in the case F = RΨ; see section 6.3.

5 The inertia action on RΨ

Elaborating an argument due to Gaitsgory [Ga], in this section we prove that RΨ carries a purely unipotent action of the inertia group. This has two important consequences. First, it implies that the Jordan-Hölder series for RΨ in the category P^{B}(Fl ×_{s} η, \overline{ℚ}_{ℓ}) is determined by that of its image in the category P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) (and that the choice of lift of Frobenius is immaterial for the latter). Secondly, it shows that the semi-simple trace of Frobenius and the usual trace of Frobenius agree on these nearby cycles, so that the Kottwitz conjecture can effectively be used to derive information about the Jordan-Hölder series in this latter category. (See sections 6,7 and 9.)

5.1 Unipotent/non-unipotent decompositions

24 5 THE INERTIA ACTION ON RΨ

Let (S, η, s) be a Henselian trait, and let (S̄, η̄, s̄) be the trait constructed by letting η̄ be a separable closure of η, S̄ the normalization of S in η̄, and s̄ the corresponding closed point. We assume the residue field k(s) = 𝔽_{q}, q a power of a prime p. Write Γ = Gal(η̄/η), Γ^{s} = Gal(s̄/s), and let Γ_{0} denote the inertia subgroup, i.e., the kernel of the canonical homomorphism Γ → Γ^{s}. For a fixed prime l ≠ p, there is a canonical surjective homomorphism

t_{l} : Γ_{0} → ℤ_{l}(1).

Let ℚ_{l}-Mod denote the category of finite dimensional ℚ_{l}-vector spaces. We consider the abelian category Rep(Γ, ℚ_{l}) of continuous finite-dimensional l-adic representations (ρ, V)

ρ : Γ → GL(V).

By a theorem of Grothendieck (cf. [ST]), the restricted representation ρ(Γ_{0}) is quasi-unipotent i.e., there exists a finite-index subgroup Γ_{1} of Γ_{0} which acts unipotently on V. There exists then a unique nilpotent morphism, the logarithm of the unipotent part of ρ

N : V (1) → V

characterized by the following property: for all g ∈ Γ_{1}, we have

ρ(g) = exp(Nt_{l}(g)).

We make a choice of geometric Frobenius Φ lifting the automorphism a ↦ a^{1/q} with respect to the homomorphism Gal(η̄/η) → Gal(s̄/s). Then ρ uniquely determines a representation ρ̃: ⟨Φ⟩ ⋉ Γ_{0} → GL(V), trivial on some finite-index subgroup Γ_{1} of Γ_{0}, by the formula

ρ(Φ^{n}σ) = ρ̃(Φ^{n}σ)exp(Nt_{l}(σ))

for n ∈ ℤ and σ ∈ Γ_{0} (cf. [Del]); moreover we have the relation ρ̃(Φ^{n}σ)Nρ̃(Φ^{n}σ)^{−1} = q^{−n}N.

We can define an idempotent endomorphism γ = γ_{V} ∈ End_{ℚ_{ℓ}}(V) by

γ = |Γ_{0}/Γ_{1}|^{−1} ∑_{g∈Γ_{0}/Γ_{1}} ρ̃(g).

Clearly this is independent of the choice of coset representatives for Γ_{0}/Γ_{1} as well as the choice of finite-index subgroup Γ_{1} ⊂ Γ_{0} on which ρ̃ is trivial. Using this it is easy to see that γ commutes with ρ̃(⟨Φ⟩ ⋉ Γ_{0}) and N, hence also with ρ(Γ). We obviously have

γ^{2} = γ.

We note that if f : V → W is a morphism in Rep(Γ, ℚ_{l}), then f ∘ γ_{V} = γ_{W} ∘ f.

We define the unipotent and non-unipotent subrepresentations of V as follows: V^{un} (resp. V^{non−un}) is the largest Γ-subrepresentation such that, as a representation of Γ_{0}, all of its irreducible subquotients are trivial (resp. non-trivial). It is easy to see that these exist, and that V^{un} ∩ V^{non−un} = (0).

We denote by Rep(Γ, ℚ_{l})^{un} (resp. Rep(Γ, ℚ_{l})^{non−un}) the maximal full subcategory of Rep(Γ, ℚ_{l}) whose objects are all unipotent (resp. non-unipotent).

Lemma 5.1 (1) For every l-adic representation (ρ, V) of Γ, there is a canon-

ical decomposition

V = V^{un} ⊕ V^{non−un}.

Moreover, any morphism f : V → W in Rep(Γ, ℚ_{l}) decomposes canonically as f = f^{un} ⊕ f^{non−un} : V^{un} ⊕ V^{non−un} → W^{un} ⊕ W^{non−un}. This gives rise to a decomposition of abelian categories

Rep(Γ, ℚ_{l}) = Rep(Γ, ℚ_{l})^{un} ⊕ Rep(Γ, ℚ_{l})^{non−un}.

(2) The endomorphism γ is ρ(Γ)-equivariant. Furthermore, we have V^{un} = ker(1 − γ) = im(γ) and V^{non−un} = ker(γ) = im(1 − γ).

(3) V is unipotent (resp. non-unipotent) if and only if 1 − γ (resp. γ) acts by zero on V.

Proof. (1). We only need to show that V has some decomposition V = V_{1} ⊕ V_{2}, where V_{1} is unipotent and V_{2} is non-unipotent.

Since the restriction of ρ̃ to Γ_{0} is semi-simple, we may consider the decomposition of V according to its isotypical components:

V = ⨁_{τ} V_{τ} .

For any fixed σ, the automorphism exp(Nt_{l}(σ)) commutes with every ρ̃(σ′), σ′ ∈ Γ_{0}, hence preserves each component V_{τ}. Thus ρ(σ) preserves each component V_{τ}, for σ ∈ Γ_{0}. On the other hand, it is easily seen that ρ(Φ) preserves the isotypical component V_{𝕀} corresponding to the trivial representation, while permuting the components V_{τ} with τ ≠ 𝕀. Since the Weil group ⟨Φ⟩ ⋉ Γ_{0} is dense in Γ, it follows that V_{1} = V_{𝕀} and V_{2} = ⨁_{τ≠𝕀} V_{τ} have the required properties.

(2). We have already noted that γ is ρ(Γ)-equivariant. The exact sequence

\xymatrix{0 \ar[r] & V^{ρ̃(Γ_{0})} \ar[r]^{i} & V \ar[r]^{p} & V/V^{ρ̃(Γ_{0})} \ar[r] & 0}

splits in the category of ρ(Γ)-modules; in fact we have γ ∘i = id and p∘(1−γ) = id. This shows that

V^{un} = V^{ρ̃(Γ_{0})} = im(γ)
V^{non−un} ≅ V/V^{ρ̃(Γ_{0})} ≅ ker(γ).

(3). This follows from (2). □

Let π_{1} denote a pro-finite group (e.g. the algebraic fundamental group of a locally Noetherian scheme at a geometric point). Let Rep(Γ, π_{1}, ℚ_{l}) denote the category whose objects V are ℚ_{l}-modules equipped with continuous commuting actions of π_{1} and Γ and whose morphisms are compatible with the actions. Then Lemma 5.1 implies that this category decomposes into unipotent and non-unipotent subcategories.

Lemma 5.2 There is a canonical decomposition of abelian categories

Rep(Γ, π_{1}, ℚ_{l}) = Rep(Γ, π_{1}, ℚ_{l})^{un} ⊕ Rep(Γ, π_{1}, ℚ_{l})^{non−un}.

Similarly, there is a decomposition

Rep(Γ_{0}, π_{1}, ℚ_{l}) = Rep(Γ_{0}, π_{1}, ℚ_{l})^{un} ⊕ Rep(Γ_{0}, π_{1}, ℚ_{l})^{non−un},

and the restriction functor Rep(Γ, π_{1}, ℚ_{l}) → Rep(Γ_{0}, π_{1}, ℚ_{l}) is compatible with the decompositions. □

Let X denote a locally Noetherian scheme. Let LS(X, ℚ_{l}) denote the category of locally constant and constructible ℚ_{l}-sheaves on X, i.e., the l-adic local systems on X (see [SGA 5], Ex. VI §1.4). An object in this category is a surjective projective system F = (F_{n})_{n∈ℕ} where F_{n} is a finite and locally constant ℤ/l^{n}ℤsheaf on X_{ét}. If X is connected, then by loc.cit. §1.4.2, there is an equivalence of categories

LS(X, ℚ_{l}) ≅ Rep(π_{1}(X, a), ℚ_{l}),

given by the functor F_{a} : (F_{n})_{n} ↦ \underleftarrow{lim}(F_{n})_{a} ⊗_{ℤ_{ℓ}} ℚ_{ℓ}, where G_{a} is the fiber of a sheaf G at a geometric point a of X. (The choice of a is essentially irrelevant, since X is connected.)

Now suppose X is a scheme of finite-type over s = Spec(𝔽_{q}), and denote the geometric special fiber by Xs̄ = X ⊗_{𝔽_{q}} 𝔽̄_{q}. If Xs̄ is connected, we denote its fundamental group by π̄_{1} = π_{1}(Xs̄).

We will define several categories attached to X/s.

- LS(X ×_{s} η, ℚ_{l}) : objects are objects in LS(Xs̄, ℚ_{l}) equipped with a continuous action of Γ compatible with the action of Γ on Xs̄ (the quotient Γ^{s} of Γ acts on Xs̄ by transport of structure). In other words, (F_{n})_{n} is an object of this category if for each g ∈ Γ there is an isomorphism of projective systems

ϕ_{g} : ḡ_{∗}F_{n} ≅ F_{n}

such that ϕ_{gh} = ϕ_{g} ∘ ḡ_{∗}(ϕ_{h}). Here ḡ denotes the image of g under the projection Γ → Γ^{s}. (Note: The “continuity” condition is that for each quasi-compact étale morphism U → X_{s} and corresponding base change Ū → Xs̄, the action of Γ on the (finite) set F_{n}(Ū) is continous for each n (cf. [SGA 7], Ex. XIII).)

Let LS(X ×_{s} η^{nr}, ℚ_{l}) denote the category consisting of local systems with an action of Γ_{0} rather than a compatible action of the group Γ. If Xs̄ is connected, then taking stalks G ↦ G_{a} gives an equivalence of categories

LS(X ×_{s} η^{nr}, ℚ_{l}) ≅ Rep(Γ_{0}, π̄_{1}, ℚ_{l}).

(Remark: This means that to define the notions of unipotent and nonunipotent for objects on the left hand side, we should consider their stalks, where such notions are already defined; cf. Lemma 5.2.)

- Sh_{c}(X ×_{s} η, ℚ_{l}) : Let Sh_{c}(Xs̄, ℚ_{l}) denote the abelian category consisting of ℚ_{l}-sheaves (F_{n})_{n} on Xs̄ where each F_{n} is finite and constructible.

We define Sh_{c}(X×_{s}η, ℚ_{l}): the objects are objects of Sh_{c}(Xs̄, ℚ_{l}) equipped with a continuous action of Γ, compatible with the action of Γ on Xs̄, in the sense defined above. The morphisms are compatible with the action of Γ. Continuity has the same meaning as in LS(X ×_{s} η, ℚ_{l}) above.

The category Sh_{c}(X×_{s}η^{nr}, ℚ_{l}) is defined analogously to LS(X×_{s}η^{nr}, ℚ_{l}).

- P (X ×_{s} η, ℚ_{l}) : the objects are middle perverse sheaves on Xs̄ equipped with a continuous action of Γ compatible with its action on Xs̄. (Here continuity is tested on cohomology sheaves.) Morphisms are compatible with the action of Γ. This category is naturally a full subcategory of the category D_{c}^{b}(X ×_{s} η, ℚ_{l}) defined below.

The category P (X ×_{s} η^{nr}, ℚ_{l}) is defined analogously to LS(X ×_{s} η^{nr}, ℚ_{l}).

- D_{c}^{b}(X ×_{s} η, ℚ_{l}) : the objects are objects of the “derived” category D_{c}^{b}(Xs̄, ℚ_{l}) equipped with a continuous action of Γ compatible with its action on Xs̄. (Continuity is tested on the cohomology sheaves.) The morphisms are compatible with the action of Γ.

The category D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}) is analogous to LS(X ×_{s} η^{nr}, ℚ_{l}).

The following lemma extends certain properties of Rep(Γ_{0}, ℚ_{l}) to each of the analogous categories above.

Lemma 5.3 Let M be an object of one of the categories LS(X ×_{s} η^{nr}, ℚ_{l}), Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}), or D_{c}^{b}(X ×_{s} η^{nr}, Q_{l}). Then

(1) The action of ρ(Γ_{0}) on M is quasi-unipotent: there exists a finite-index subgroup Γ_{1} ⊂ Γ_{0} such that (ρ(g) − 1) acts nilpotently on M, for every g ∈ Γ_{1}.

(2) There is a unique nilpotent morphism N : M(1) → M characterized by

the equality

ρ(g) = exp(Nt_{l}(g))

for every g ∈ Γ_{1}.

(3) There is a unique homomorphism ρ̃: Γ_{0} → Aut(M) which satisfies

(a) ρ(σ) = ρ̃(σ)exp(Nt_{l}(σ)), for σ ∈ Γ_{0},

(b) ρ̃ is trivial on a finite-index subgroup of Γ_{0},

(c) ρ̃(σ)Nρ̃(σ)^{−1} = N, for σ ∈ Γ_{0}.

This is closely related to the next result, whose proof works along the same lines. To state it we need a suitable notion of “unipotent” and “non-unipotent” object for the categories above.

Definition 5.4 (1) Let M be an object of LS(X ×_{s} η, ℚ_{l}) or Sh_{c}(X ×_{s} η, ℚ_{l}). We say M is unipotent (resp. non-unipotent) if for every geometric point x̄ ∈ X_{s}(𝔽̄_{q}), the stalk Mx̄ is a unipotent (resp. non-unipotent) object of Rep(Γ_{0}, ℚ_{l}).

(2) Let M be an object of P (X×_{s}η, ℚ_{l}) or D_{c}^{b}(X×_{s}η, ℚ_{l}). We say M is unipotent (resp. non-unipotent) if every cohomology sheaf H^{i}M is a unipotent (resp. non-unipotent) object of Sh_{c}(X ×_{s} η, ℚ_{l}).

Remark: In the same way we can define the notions of unipotent and nonunipotent for objects of LS(X ×_{s} η^{nr}, ℚ_{l}), Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}), etc. It is clear that M ∈ LS(X ×_{s} η, ℚ_{l}) etc. is unipotent (resp. non-unipotent) if and only if it has unipotent (resp. non-unipotent) image under the forgetful functor LS(X ×_{s} η, ℚ_{l}) → LS(X ×_{s} η^{nr}, ℚ_{l}) etc.

Lemma 5.5 (1) Let C be one of the categories LS(X ×_{s} η^{nr}, ℚ_{l}), Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}), or D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}). Let M be an object of C. Define the idem-

potent γ = γ_{M} ∈ End_{C}(M) by

γ = |Γ_{0}/Γ_{1}|^{−1} ∑_{g∈Γ_{0}/Γ_{1}} ρ̃(g)

where ρ̃ is as above and Γ_{1} ⊂ Γ_{0} is any finite-index subgroup of Γ_{0} on which ρ̃ is trivial. Then M is unipotent (resp. non-unipotent) if and only if 1 − γ_{M} (resp. γ_{M}) acts by zero on M.

(2) If M is an object of one of the abelian categories LS(X ×_{s} η^{nr}, ℚ_{l}), Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}), or P (X ×_{s} η^{nr}, ℚ_{l}) then the canonical decomposition

M = ker(1 − γ_{M}) ⊕ ker(γ_{M})

is a canonical decomposition of M into unipotent and non-unipotent direct factors. The same holds for the categories LS(X×_{s}η, ℚ_{ℓ}), Sh_{c}(X×_{s}η, ℚ_{ℓ}), and P (X ×_{s} η, ℚ_{ℓ}).

Proof of Lemmas 5.3 and 5.5.

To begin, we assume M ∈ D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}), the most general category in Lemma 5.3.

Once it is known that ρ(Γ_{0}) acts quasi-unipotently on M, the existence and uniqueness of N follow in general from the same argument used in the category Rep(Γ_{0}, ℚ_{l}). Indeed, let P_{l} = ker(t_{l}) and let Γ_{1} ⊂ Γ_{0} denote a finite-index subgroup such that ρ(Γ_{1}) acts unipotently on M. One shows that ρ(Γ_{1} ∩ P_{l}) = id_{M} (this can be tested on cohomology stalks, effectively reducing to the case M ∈ Rep(Γ_{0}, ℚ_{l}) where it is already known), and thence that ρ|Γ_{1} factors through t_{l}; then define N : M(1) → M by Nt_{l}(σ) = log(ρ(σ)) for all σ ∈ Γ_{1}.

Moreover, given N we then define ρ̃ by the equality

ρ̃(σ) = ρ(σ)exp(−Nt_{l}(σ))

for σ ∈ Γ_{0}. The identities proving that ρ̃ is a homomorphism and the equality in Lemma 5.3 (3(c)) reduce to the case M ∈ Rep(Γ_{0}, ℚ_{l}). Indeed, passing to cohomology sheaves, we may assume M ∈ Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}). Then the constructibility of M allows us to view M as the result of “gluing” a local system on an open subscheme with a constructible sheaf supported on a proper closed subscheme (for the precise statement of gluing, see for example [Mi], II §3). By Noetherian induction therefore, we may assume M ∈ LS(X ×_{s} η^{nr}, ℚ_{l}). Passing to the finitely many connected components of X, we are reduced to M ∈ Rep(Γ_{0}, π̄_{1}, ℚ_{l}), where the desired identities are already known.

To prove that ρ̃ is trivial on a finite-index subgroup Γ_{1} ⊂ Γ_{0}, one uses the same reduction steps, keeping in mind that for M ∈ Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}), there is a finite stratification X_{i} (i ∈ I) of X by locally closed subschemes X_{i} such that M|X_{i} ∈ LS(X_{i} ×_{s} η^{nr}, ℚ_{l}), and each X_{i} has only finitely many connected components.

To show that ρ(Γ_{0}) acts quasi-unipotently on M is similar. This completes the proof of Lemma 5.3.

Now again suppose M is an object of D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}). We wish to prove M is unipotent (resp. non-unipotent) if and only if 1 − γ_{M} (resp. γ_{M}) acts by zero on M. Passing to cohomology sheaves and using the obvious identity

γ_{H^{i}M} = H^{i}(γ_{M})

we can assume M ∈ Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}). Taking stalks and using the identity

γ_{i∗ ; x̄M} = i∗ ; x̄(γ_{M}),

where ix̄ : x̄ → X_{s} is the embedding of a geometric point in X_{s}(𝔽̄_{q}), we can assume that M ∈ Rep(Γ_{0}, ℚ_{l}), where the result follows from Lemma 5.1. This proves Lemma 5.5 part (1).

Concerning part (2), if M belongs to one of the abelian categories LS(X ×_{s} η^{nr}, ℚ_{l}), Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}), or P (X ×_{s} η^{nr}, ℚ_{l}), then the idempotent endomorphism γ_{M} gives rise to the required decomposition. This proves part (2), and thus Lemma 5.5. □

Remark: If C is one of the categories in Lemma 5.5 (2), there is a decomposition of abelian categories

C = C^{un} ⊕ C^{non−un},

and we denote the corresponding components of an object M by M^{un} and M^{non−un}.

This argument does not prove that the category D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}) decomposes in a like manner: since it is not abelian, we cannot even define the kernel and image of γ_{M} for a general object M.

5.2 Permanence properties of unipotence and non-unipotence

Suppose f : X → Y is a finite-type morphism of finite-type S-schemes. This gives rise to the “derived” functors f_{!} : D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}) → D_{c}^{b}(Y ×_{s} η^{nr}, ℚ_{l}) and f^{∗} : D_{c}^{b}(Y ×_{s} η^{nr}, ℚ_{l}) → D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}).

Lemma 5.6 (1) Let M ∈ D_{c}^{b}(X ×_{s} η^{nr}, ℚ_{l}) be unipotent (resp. nonunipotent). Then f_{!}M ∈ D_{c}^{b}(Y ×_{s} η^{nr}, ℚ_{l}) is unipotent (resp. nonunipotent).

(2) Let M′ ∈ D_{c}^{b}(Y ×_{s} η^{nr}, ℚ_{l}), and suppose fs̄ : Xs̄ → Ys̄ is surjective. Then M′ is unipotent (resp. non-unipotent) if and only if f^{∗}M′ is unipotent (resp. non-unipotent).

Proof. (1). By the local-global spectral sequence

H^{p}(f_{!}H^{q}M) ⇒ H^{p+q}f_{!}M

we are reduced to the case M ∈ Sh_{c}(X ×_{s} η^{nr}, ℚ_{l}). Taking cohomology stalks we know that f_{!}M is unipotent if and only if for every ȳ ∈ Y_{s}(𝔽̄_{q}), and every i, 1 − γ acts by zero on H_{c}^{i}(f^{−1}(ȳ), M). But as M is unipotent, 1 − γ acts by zero on M, hence on the cohomology group with coefficients in M as well. The non-unipotent case is similar.

(2). This is straightforward. □

5.3 Unipotence/non-unipotence of products and convolutions

The following facts follow easily from Lemmas 5.5 and 5.6.

Let F and G be objects of D_{c}^{b}(X ×_{s} η, ℚ_{l}) and suppose that F is unipotent. Then (F ⊠ G)^{non−un} = F ⊠ G^{non−un}.

Now let B denote the Iwahori-group scheme acting on the affine flag variety Fl = G(𝔽̄_{p}((t)))/B. If the objects F and G above are B-equivariant perverse sheaves on Fl, then we may define their twisted product F ˜⊠G and their convolution product F ⋆_{s} G (cf. [HN1]). Still under the assumption F = F^{un}, one can prove first (F ˜⊠G)^{non−un} = F ˜⊠G^{non−un}, and then (F ⋆_{s} G)^{non−un} = F ⋆_{s} G^{non−un}.

5.4 RΨ = RΨ^{un}

First, we remark that in the above discussion in 5.1-5.3, we can without any problems replace ℚ_{ℓ} by a finite extension, and can then also pass to the 2inductive limit of the categories D_{c}^{b}(X, E), E/ℚ_{ℓ} finite. Thus all the statements above hold for \overline{ℚ}_{ℓ}-adic sheaves as well.

We now come back to the affine flag variety. Fix a dominant coweight μ and let RΨ = RΨ(IC(Q_{μ})[ℓ(μ)]) ∈ P^{B}(Fl ×_{s} η, \overline{ℚ}_{ℓ}).

Theorem 5.7 (Gaitsgory) The action of Γ_{0} on the sheaf RΨ of nearby cycles is purely unipotent.

Proof. Because RΨ = RΨ^{un}⊕RΨ^{non−un} it suffices to prove that RΨ^{non−un} = 0.

Note that RΨ^{non−un} is a B-equivariant object of P (Fl×_{s}η, ℚ_{l}) (for equivariance, use Lemma 5.6). By using 5.3 we see that, as for RΨ itself, RΨ^{non−un} is central with respect to convolution of Iwahori-equivariant perverse sheaves on the affine flag variety. To show it is zero, it suffices to show that its semi-simple trace function τ_{RΨ^{non−un}} : x ↦ Tr^{ss}(Fr_{q}, RΨ_{x}^{non−un}), an element in Z(H_{I}), vanishes.

Now consider the projection π : Fl → Grass. Because the semi-simple trace provides a good sheaf-function dictionary in the sense of Grothendieck (cf. [HN1], Proposition 10), the semi-simple trace function Tr^{ss}(Fr_{q}, π_{∗}(RΨ^{non−un})) corresponds to summing Tr^{ss}(Fr_{q}, RΨ^{non−un}) along the fibers of π, i. e., we have the following equality of functions in the spherical Hecke algebra H_{K}:

Tr^{ss}(Fr_{q}, π_{∗}(RΨ^{non−un})) = Tr^{ss}(Fr_{q}, RΨ^{non−un}) ⋆ 𝕀_{K}.

By a theorem of Bernstein (cf. [L2]), − ⋆ 𝕀_{K} : Z(H_{I}) ˜→H_{K} is an isomorphism, so it suffices to prove π_{∗}(RΨ^{non−un}) = 0. But by Lemma 5.6 we have

π_{∗}(RΨ^{non−un}) = (π_{∗}RΨ)^{non−un} = (A_{μ,s})^{non−un} = 0.

Here A_{μ} is the “constant” intersection complex IC(Q_{μ})[ℓ(μ)] on the “constant family” Grass, hence it has trivial inertia action. We are done.

5.5 Semi-simple trace vs. usual trace

Proposition 5.8 The trace and the semi-simple trace on RΨ coincide:

Tr^{ss}(Fr_{q}, RΨ) = Tr(Fr_{q}, RΨ^{ss}).

Here, on the left hand side RΨ is regarded as an object of the category P^{B}(Fl×_{s} η, \overline{ℚ}_{ℓ}), and the semi-simple trace is defined as in [HN1]. On the right hand side, Fr_{q} represents a choice of lift of geometric Frobenius, and RΨ^{ss} denotes the semi-simplification of RΨ when regarded as an object of P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) (via that choice of lift).

Proof. It suffices to find a Galois-invariant filtration on RΨ such that Γ_{0} acts trivially on the subquotients. Since Γ_{0} acts unipotently on RΨ by 5.4, such a filtration is given by the kernels of powers of the operator N:

F^{i}RΨ = kerN^{i}.

32 6 APPLYING THE THEOREM TO RΨ

5.6 A different proof

Here is a sketch of a different argument for the result above: after fixing a Weil sheaf structure on RΨ, the usual trace of Frobenius satisfies the same sheaffunction dictionary as the semi-simple trace: the same proof as for semi-simple trace shows that the usual trace yields a central function in the Iwahori-Hecke algebra, characterized by

Tr(Fr_{q}, RΨ) ⋆ 𝕀_{K} = Tr(Fr_{q}, IC(Q_{μ})[ℓ(μ)]).

It follows from this that it coincides with the semi-simple trace function. This in turn implies that the action of inertia on RΨ must be purely unipotent. To prove the last statement, it is enough to show that RΨ admits a filtration such that the inertia group Γ_{0} acts trivially on the graded pieces. Consider the filtration

F^{i}RΨ = ker N^{i},

where N is the logarithm of the unipotent part of the local monodromy, as before. The inertia group acts on the F^{i}/F^{i−1} through a finite quotient, so the definition of semi-simple trace and the above remarks yields

Tr(Fr_{q}, (⨁ F^{i}/F^{i−1})^{Γ_{0}}) = Tr^{ss}(Fr_{q}, RΨ) = Tr(Fr_{q}, RΨ) = Tr(Fr_{q}, ⨁ F^{i}/F^{i−1}).

Thus the function of the (necessarily perverse) quotient (⨁ F^{i}/F^{i−1})/(⨁ F^{i}/F^{i−1})^{Γ_{0}} is zero, so we have (⨁ F^{i}/F^{i−1})^{Γ_{0}} = ⨁ F^{i}/F^{i−1}. (We have used the fact that if F ∈ P (Fl ×_{s} η, \overline{ℚ}_{ℓ}) carries a finite action of inertia, then F^{Γ_{0}} = im(γ_{F}), hence is also perverse.)

6 Applying the theorem to RΨ

Fix a dominant coweight μ and write RΨ = RΨ(IC(Q_{μ})[ℓ(μ)]) ∈ P^{B}(Fl ×_{s} η, \overline{ℚ}_{ℓ}).

6.1 RΨ as a Weil sheaf

Fixing a lift Fr_{q} ∈ Γ of the inverse of the Frobenius morphism enables us to view RΨ as a Weil-perverse sheaf. Since it is also a B-equivariant perverse sheaf, it is a mixed Weil-perverse sheaf (this also follows from the Appendix, Theorem 10.6).

The Kottwitz conjecture holds for RΨ (see section 2.7), so

Tr(Fr_{q}, RΨ) = ε_{μ}q_{μ}^{1/2} ∑_{λ≤μ} m_{μ}(λ)z_{λ}.

In particular, Tr(Fr_{q}, RΨ_{x}) ∈ ℤ[q, q^{−1}], so we can consider RΨ as an object of P_{q}^{B}(Fl, \overline{ℚ}_{ℓ}).

6.2 RΨ and property (P) 33

We will show that the nearby cycles sheaf satisfies the property (P), and hence we can apply Theorem 4.7, and get some information on the Jordan-Hölder series of RΨ as a Weil sheaf.

But furthermore, we know that the inertia group Γ_{0} acts purely unipotently on RΨ, so in this case, we get a stronger independence statement with respect to the choice of the lift Fr_{q}. Namely, we can regard the sheaves IC_{w}(−i), with trivial Γ_{0}-action, as objects of P (Fl×_{s}η, \overline{ℚ}_{ℓ}), and then the Jordan-Hölder series of RΨ ∈ P (Fl ×_{s} η, \overline{ℚ}_{ℓ}) is ’the same’ as that of RΨ as a Weil-perverse sheaf.

6.2 RΨ and property (P)

To show that RΨ satisfies the property (P), we need to prove

\overline{Tr(Fr_{q}, RΨ_{x})} = ε_{x}ε_{μ}q_{x}q_{μ}^{−1}Tr(Fr_{q}, RΨ_{x}),

because DRΨ = RΨ(ℓ(t_{μ})), i.e. Tr(Fr_{q}, DRΨ) = q_{μ}^{−1}Tr(Fr_{q}, RΨ).

Note the following general lemma. We write Q = q^{−1/2} − q^{1/2}.

Lemma 6.1 Let f ∈ ℤ[q, q^{−1}], α ∈ \frac{1}{2}ℤ such that f = q^{α}R(Q) for some R ∈ ℤ[Q]. Then all Q-powers occurring in R have the parity of 2α, and f(q^{−1}) = (−1)^{2α}q^{−2α}f(q). □

Recalling that z_{λ} = ∑_{ν∈W (λ)} Θ_{ν} and the fact that Θ_{ν} = T̃_{ν_{1}} T̃_{ν_{2}}^{−1}, where ν_{1}−ν_{2} = ν and ν_{i} is dominant, we see (using the Kottwitz conjecture) that

RΨ(x) = ε_{μ}q_{μ}^{1/2}q_{x}^{−1/2}R(Q),

where R ∈ ℕ[Q]. Thus we can apply the lemma with α = (ℓ(μ) − ℓ(x))/2, and get that RΨ satisfies (P).

6.3 Consequences of the theorem for RΨ

Since RΨ is self-dual up to a Tate twist, from Theorem 4.7 we get

Corollary 6.2 We have

∑_{i} m(RΨ, w, i)q^{i} = ε_{μ}Tr(Fr_{q}, H_{c}^{•}(Fl, RΨ ⊗ IC(B\overline{w}))).

In particular, by setting w = τ, the unique element of length 0 in the support of RΨ, we get by using the projection formula for π : Fl → Grass that

∑_{i} m(RΨ, τ, i)q^{i} = ε_{μ}Tr(Fr_{q}, H_{c}^{•}(Fl, RΨ))
= ε_{μ}Tr(Fr_{q}, H_{c}^{•}(Grass, π_{∗}RΨ))
= ε_{μ}Tr(Fr_{q}, H_{c}^{•}(Grass, IC(Q_{μ})[ℓ(μ)]))
= ∑_{i} dim IH^{2i}(Q_{μ}, \overline{ℚ}_{ℓ})q^{i}.

Corollary 6.3 The multiplicity m(RΨ, τ, i) is the 2i-th intersection Betti number of Q̄_{μ}.

Remark. This corollary was initially noticed as a purely empirical fact with the aid of a computer program, and it was the starting point of this work: the effort to explain it conceptually lead to the discovery of the more general Theorem 4.7. The corollary can also be proved, taking the Kottwitz conjecture into account, by purely combinatorial considerations in the Iwahori-Hecke algebra H (recall that the traces of Frobenius and the multiplicities are related by a base change in H). But it seems impossible to prove anything about the multiplicities for other w’s in that way.

More generally, by applying section 4.6.1 to F = RΨ, we get

Corollary 6.4 Suppose w ∈ Adm(μ) belongs to (W̃/W)_{min}. Then the multiplicity function is given by

∑_{i} m(RΨ, w, i)q^{i} = Tr(Fr_{q}, H_{c}^{•}(Q̄_{μ}, IC(Q̄_{μ}) ⊗ IC(Q\overline{π(w)}))).

7 Wakimoto sheaves

7.1 Definition and properties

We have a convolution product

⋆: P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) × P_{Weil}^{B}(Fl, \overline{ℚ}_{ℓ}) ⟶ D_{c,Weil}^{b}(Fl, \overline{ℚ}_{ℓ}),

see [L3], [HN1]. Using the usual sheaf-function dictionary, it is easy to see that convolution of sheaves corresponds to multiplication of their functions in the Hecke algebra.

Definition 7.1 The Wakimoto sheaf associated to v, w ∈ W̃ is the sheaf M_{v,w} := j_{v!}\overline{ℚ}_{ℓ}[ℓ(v)] ⋆ j_{w∗}\overline{ℚ}_{ℓ}[ℓ(w)].

Note that, as usual, j_{v!} and j_{w∗} denote the derived functors. The function in the Iwahori-Hecke algebra H corresponding to M_{v,w} is ε_{v}ε_{w}q_{w}T_{v}T_{w^{−1}}^{−1} = ε_{v}ε_{w}q_{v}^{1/2}q_{w}^{1/2}T̃_{v}T̃_{w^{−1}}^{−1}. As usual, we write T̃_{x} = q_{x}^{−1/2}T_{x}.

Proposition 7.2 Properties of Wakimoto sheaves

(1) M_{v,w} is perverse.

(2) M_{v,w} satisfies (P), with d = ℓ(v) + ℓ(w).

7.2 The elements in the Hecke algebra corresponding to Wakimoto sheaves35

Proof. (1) This result is due to Mirkovic. See [HP] Prop. 6.2, or [AB] Thm. 5.

(2) The function corresponding to the dual of M_{v,w} is ε_{v}ε_{w}q_{w}^{−1}T_{w}T_{v^{−1}}^{−1} = ε_{v}ε_{w}q_{w}^{−1/2}q_{v}^{−1/2}T̃_{w}T̃_{v^{−1}}^{−1}. We have

DM_{v,w}(y) = ε_{v}ε_{w}q_{w}^{−1/2}q_{v}^{−1/2}q_{y}^{−1/2}R_{y,v}^{w}(Q)

where R_{y,v}^{w} ∈ ℕ[Q] is such that Q^{i} occurs only for i ≡ ℓ(w) + ℓ(v) + ℓ(y) mod 2. (Use, for example, Lemma 6.1.) Thus,

\overline{DM_{v,w}(y)} = ε_{v}ε_{w}ε_{y}q_{w}q_{v}q_{y}DM_{v,w}(y),

which proves property (P) with d = ℓ(v) + ℓ(w). □

The proposition shows that there is a large class of sheaves which satisfy the property (P).

Proposition 7.3 Let F ∈ P_{√\overline{q}}^{B}(Fl, \overline{ℚ}_{ℓ}) be a sheaf which has a filtration whose graded pieces are of the form M_{v_{i},w_{i}}(−α_{i}), where α_{i} ∈ ℤ is such that 2α_{i}+ℓ(v_{i})+ ℓ(w_{i}) is independent of i. Then F satisfies (P), with d = ℓ(v_{i}) + ℓ(w_{i}) + 2α_{i}.

Proof. The function corresponding to F is

∑_{i} q^{α_{i}}q_{w_{i}}ε_{v_{i}}ε_{w_{i}}T_{v_{i}}T_{w_{i}^{−1}}^{−1},

and the function of the dual of F is obtained by applying \overline{·}. A computation similar to the one above yields the result. □

Remark. In their recent paper [AB], Arkhipov and Bezrukavnikov show that the sheaf of nearby cycles RΨ has a filtration whose graded pieces are Wakimoto sheaves.

Their theorem relies on the sheaf of nearby cycles being central, so that we do not get a proof of (P) for RΨ which does not use the Kottwitz conjecture.

7.2 The elements in the Hecke algebra corresponding to Wakimoto sheaves

Fix v, w ∈ W_{aff}, and suppose w = s_{1} ⋯ s_{r} is reduced. There are unique polynomials R_{x,w}^{v}(Q) in the parameter Q := q^{−1/2} − q^{1/2} and with non-negative integral coefficients, such that

T̃_{v}T̃_{w^{−1}}^{−1} = ∑_{x≤v|w} R_{x,w}^{v}(Q)T̃_{x}.

Here the notation x ≤ v|w means by definition that x is the terminal element of a v-distinguished subexpression with respect to w (see section 5 of [H1]). Recall that an r+1-tuple [σ_{0}, …, σ_{r}] is v-distinguished if σ_{0} = v, and if for each j ≥ 1, 1. σ_{j} ∈ {σ_{j−1}, σ_{j−1}s_{j}}; and

2. σ_{j} = σ_{j−1} only if σ_{j−1} < σ_{j−1}s_{j}.

We are going to prove a formula for R_{x,w}^{v}(Q). We need first some more definitions.

For a v-distinguished subexpression \underline{σ} = [σ_{0}, …, σ_{r}], we set π(\underline{σ}) = σ_{r}. Also define

D(x) = \{\underline{σ} | π(\underline{σ}) = x\}.

Further define integers n(\underline{σ}) = |\{j | σ_{j−1} = σ_{j}\}|, and m(\underline{σ}) = |\{j | σ_{j−1} > σ_{j}\}|. We have the following easy lemma.

Lemma 7.4 If \underline{σ} is v-distinguished with respect to w, then

ℓ(π(\underline{σ})) = ℓ(v) + ℓ(w) − n(\underline{σ}) − 2m(\underline{σ}).

Now write w′ = s_{1} ⋯ s_{r−1}, so that w′ = ws_{r} < w. The equality

∑_{y≤v|w′} R_{y,w′}^{v}(Q)T̃_{y}(T̃_{s_{r}} + Q) = ∑_{x≤v|w} R_{x,w}^{v}(Q)T̃_{x}

give rise to recurrence relations for the polynomials R_{x,w}^{v}. Using these, we can prove the following proposition by induction on ℓ(w).

Proposition 7.5 If x ≤ v|w, then

R_{x,w}^{v}(Q) = ∑_{\underline{σ}∈D(x)} Q^{n(\underline{σ})} = ∑_{\underline{σ}∈D(x)} Q^{ℓ(v)+ℓ(w)−ℓ(x)−2m(\underline{σ})}.

Taking the parity of the exponents into account, this proposition gives a direct verification of the conclusion of Lemma 6.1 for Wakimoto sheaves.

Proof. The proof is by induction on ℓ(w), and uses case-by-case analysis. Write w′ as above and write s_{r} = s. We consider the set D′(y) of v-distinguished subexpressions with respect to w′ which end with y. Integers n and m can be attached to each of these in the same way as before.

Note that x ≤ v|w implies that xs ≤ v|w′ or x ≤ v|w′, or both.

Case 1: x ≤ v|w, x ≤ v|w′, and xs ≰ v|w′.

(a) x < xs. Then every \underline{σ} ∈ D(x) is of the form [?, x, x]. Given such, write \underline{σ}′ = [?, x] (the first part of \underline{σ}). We see n(\underline{σ}) = n(\underline{σ}′) + 1, m(\underline{σ}) = m(\underline{σ}′), and R_{x,w}^{v} = QR_{x,w′}^{v}. Moreover, we have a bijection of sets S : D(x) ˜→D′(x) given by \underline{σ} ↦ \underline{σ}′.

(b) xs < x. This cannot happen, since the only possible \underline{σ}′ is of form [?, x], and \underline{σ} cannot have form [?, x, x], since then we would need to have x < xs (by definition of v-distinguished).

7.3 The situation at q = 1 37

Case 2: x ≤ v|w, x ≰ v|w′, and xs ≤ v|w′.

(a) x < xs. Then every \underline{σ} is of form [?, xs, x]; set \underline{σ}′ = [?, xs]. We have n(\underline{σ}) = n(\underline{σ}′), and R_{x,w}^{v} = R_{xs,w′}^{v}. Also, S : D(x) ˜→D′(xs) by \underline{σ} ↦ \underline{σ}′.

(b) xs < x. Then \underline{σ} is of form [?, xs, x]; let \underline{σ}′ = [?, xs]. We have n(\underline{σ}) = n(\underline{σ}′), and R_{x,w}^{v} = R_{xs,ws}^{v}, and S : D(x) ˜→D′(xs) by \underline{σ} ↦ \underline{σ}′.

Case 3: x ≤ v|w, x ≤ v|w′, and xs ≤ v|w′.

(a) x < xs. We define A(x) = \{\underline{σ} is of the form [?, xs, x]\}, and B(x) = \{\underline{σ} is of the form [?, x, x]\}. Note that D(x) = A(x) ∐ B(x).

For \underline{σ} = [?, xs, x] ∈ A(x), let \underline{σ}′ = [?, xs]. The map \underline{σ} ↦ \underline{σ}′ gives A(x) ˜→D′(xs), and n(\underline{σ}) = n(\underline{σ}′).

For \underline{σ} = [?, x, x] ∈ B(x), let \underline{σ}′ = [?, x]. The map \underline{σ} ↦ \underline{σ}′ gives B(x) ˜→D′(x), and n(\underline{σ}) = n(\underline{σ}′) + 1.

In either case, we have R_{x,w}^{v} = QR_{x,ws}^{v} + R_{xs,ws}^{v}.

(b) xs < x. Then \underline{σ} = [?, xs, x], and setting \underline{σ}′ = [?, xs], we have n(\underline{σ}) = n(\underline{σ}′), S : D(x) ˜→D′(xs). We have R_{x.w}^{v} = R_{xs,ws}^{v}.

The proposition now follows easily by using the induction hypothesis and by taking into account the various cases. □

In general it is difficult to say anything really concrete concerning the polynomial R_{x,w}^{v}(Q): even its degree is mysterious. However, suppose T̃_{v}T̃_{w^{−1}}^{−1} has a minimal expression, i.e., suppose we can write

T̃_{v}T̃_{w^{−1}}^{−1} = T̃_{t_{1}}^{ϵ_{1}} ⋯ T̃_{t_{k}}^{ϵ_{k}},

where t_{1} ⋯ t_{k} = vw is a reduced expression, and ϵ_{i} ∈ {1, −1}, for each i. Letting c_{x} denote the coefficient of T̃_{x} in the expansion of such a minimal expression, one can easily prove by induction on k that the Q-degree of c_{x} is bounded above:

deg_{Q}c_{x} ≤ k − ℓ(x).

We get the following consequence, which we record for later use.

Proposition 7.6 Suppose T̃_{v}T̃_{w^{−1}}^{−1} has a minimal expression. Then

deg_{Q}R_{x,w}^{v}(Q) ≤ ℓ(vw) − ℓ(x).

7.3 The situation at q = 1

Recall from section 4.2 that the elements C′′_{w} = Tr(Fr_{q}, IC_{w}) = Tr(Fr_{q}, IC(Bw̄)[ℓ(w)]) form a basis of the Iwahori-Hecke algebra H. Consider the algebra homomorphism H → ℤ[W̃] given by specializing q^{1/2} = 1. We will denote the image of C′′_{w} by C′′_{w}|, etc. The elements T_{w} map to the usual basis of the group ring ℤ[W̃], and the elements C′′_{w} map to another basis. In fact

C′′_{w}| = ∑_{y≤w} ε_{w}P_{y,w}(1)T_{y}|.

We will identify T_{y}| with y in what follows.

Now fix a coweight λ, and write λ = λ_{1} − λ_{2}, where both λ_{i} are dominant. We have

ε_{λ}q_{λ}^{1/2}Θ_{λ} = ε_{λ}q_{λ}^{1/2}T̃_{λ_{1}} T̃_{λ_{2}}^{−1} = ∑_{w} a_{w}C′′_{w}.

Since the Wakimoto sheaf giving rise to this function is perverse and Bequivariant, we see that a_{w} ∈ ℕ[q^{1/2}, q^{−1/2}] for every w, and thus a_{w} ≠ 0 if and only if a_{w}(1) ≠ 0. We call the set {w | a_{w} ≠ 0} the IC-support of ε_{λ}q_{λ}^{1/2}Θ_{λ}.

Applying the homomorphism x ↦ x| to ε_{λ}q_{λ}^{1/2}T̃_{λ_{1}} T̃_{λ_{2}}^{−1} gives

ε_{λ}t_{λ_{1}}t_{−λ_{2}} = ε_{λ}t_{λ} = ∑_{w≤t_{λ}} Q_{w,t_{λ}}(1)C′′_{w}|.

We thus see that

a_{w}(1) = Q_{w,t_{λ}}(1),

and that the IC-support of ε_{λ}q_{λ}^{1/2}Θ_{λ} is precisely the set {w ≤ t_{λ}}.

Using now the geometric interpretation of inverse Kazhdan-Lusztig polynomials in the affine case (cf. Theorem 3.5), we get the following proposition.

Proposition 7.7 The IC-support of the function ε_{λ}q_{λ}^{1/2}Θ_{λ} is precisely the set {w ≤ t_{λ}}. For each such w, the sum of the multiplicities ∑_{i} m(λ, w, i) is equal to the intersection Euler characteristic of the space

B_{t_{λ}} ∩ B\overline{w}.

8 Applications to Shimura varieties

8.1 Relating nearby cycles on local models and Shimura varieties with Iwahori level structure

In certain cases, the singularities occurring in the bad reduction of Shimura varieties of PEL type with Iwahori level structure are equivalent to the singularities of one of the schemes M_{μ}/ℤ_{p} that we were considering. More precisely, for minuscule μ, M_{μ} is a local model in the sense of Rapoport and Zink [RZ2] of a certain Shimura variety.

Consider a Shimura variety Sh = Sh_{K}(G, X) of PEL type, with Iwahori level structure at p. (The same methods should yield analogous results for arbitrary parahoric level structure.) Denote by E its reflex field. Since Shimura varieties of PEL type are moduli spaces of abelian varieties with additional structure, there is a natural way to define a model over the ring of integers of the completion of E at a prime 𝔭 over p (see [Ko], [RZ2]).

Furthermore, Rapoport and Zink have defined a so-called local model of the Shimura variety: a scheme defined in terms of linear algebra, which has ’the same’ singularities as the model of the Shimura variety.

Now assume that G_{ℚ_{p}} is isomorphic to GL_{n,ℚ_{p}} × 𝔾_{m,ℚ_{p}} or GSp_{2g,ℚ_{p}}.

The Shimura datum gives rise to a minuscule cocharacter μ of GL_{n,ℚ_{p}} or GSp_{2g,ℚ_{p}}, respectively. The functorial description of the local model shows that it can be embedded into the deformation from Grass_{ℚ_{p}} to Fl_{𝔽_{p}} à la Laumon (see [HN1]), with generic fibre Q_{μ}. Since the local model is flat [G1], [G2], it coincides with the scheme-theoretic closure M_{μ} of Q_{μ} in this deformation.

The relation between the model of the Shimura variety over ℤ_{p} and its local model is given by a diagram

\xymatrix{Sh & M \ar[l]_{φ} \ar[r]^{ψ} & M^{loc}}

of O_{E}-schemes, where φ is a torsor under a smooth affine group scheme, and ψ is smooth. The fibres of φ are geometrically connected (more precisely, this holds for the restriction of φ to any geometric connected component of M). One can show that étale-locally around each point of the special fibre, the model of the Shimura variety and the local model are isomorphic. (See [RZ2].)

The stratification of the special fibre of M^{loc} induces stratifications of the special fibres of M and Sh: the inverse images under ψ of strata in M_{𝔽_{p}}^{loc} define a stratification of M, and there is a stratification of Sh such that the inverse images of strata under φ are the strata in M. For details see [GN]. Following loc.cit., we call the resulting stratification on Sh_{𝔽_{p}} the Kottwitz-Rapoport stratification.

The smooth base change theorem implies that the nearby cycles functor commutes with pull-back under smooth morphisms, so we have (up to shifts by the dimensions)

φ^{∗}RΨ_{Sh} ≅ RΨ_{M} ≅ ψ^{∗}RΨ_{M^{loc}}.

Furthermore, the intermediate extension commutes with pull-back under smooth morphisms (see the proof of theorem III.11.3 in [KW]), which implies that

φ^{∗}IC_{Sh,w} ≅ IC_{M,w} ≅ ψ^{∗}IC_{M^{loc},w}^{3}.

(Again, this is up to a shift by the dimensions of the strata: these intersection complexes are normalized so that they are perverse.)

3As in the introduction IC_{Sh,w} and IC_{M,w} should be interpreted as direct sums of intersection complexes over geometric connected components of Sh and M.

So we see that (since ϕ and ψ have the same relative dimension)

φ^{∗}RΨ_{Sh}^{ss} ≅ φ^{∗} (⨁_{w} ⨁_{i} IC_{Sh,w}(−i)^{m(w,i)}) ,

and because φ (restricted to any geometrically connected component of M) is smooth with geometrically connected fibres, this yields

RΨ_{Sh}^{ss} ≅ ⨁_{w} ⨁_{i} IC_{Sh,w}(−i)^{m(w,i)}.

Hence the sheaf of nearby cycles on the Shimura variety decomposes in the same way as the sheaf of nearby cycles on the local model.

8.2 The weight spectral sequence

By virtue of the Appendix, Theorem 10.1, the nearby cycles RΨ := RΨ_{Sh} is a mixed perverse sheaf on the special fiber of Sh. Therefore, RΨ possesses a weight filtration ([BBD], §5), that is, an increasing filtration W such that each graded piece

gr_{i}^{W} := gr_{i}^{W} (RΨ)

is a pure perverse sheaf of weight i. Now assume that Sh is proper over O_{E} (here E denotes the completion of the reflex field at a prime above p). Then we have a canonical isomorphism of Galois modules

H^{i}(Sh_{\overline{𝔽}_{p}}, RΨ) = H^{i}(Sh_{\overline{ℚ}_{p}}, \overline{ℚ}_{ℓ}[ℓ(μ)]),

and the weight filtration on RΨ yields the weight spectral sequence

W E_{1}^{pq} = H^{p+q}(Sh_{\overline{𝔽}_{p}}, gr_{−p}^{W}) ⟹ H^{p+q}(Sh_{\overline{ℚ}_{p}}, \overline{ℚ}_{ℓ}[ℓ(μ)]).

This spectral sequence degenerates in E_{2}, and abuts to the weight filtration on the cohomology group on the right hand side. Deligne’s monodromy-weight conjecture asserts that the weight filtration on the right hand side is the same as the monodromy filtration defined with the help of the inertia action.

One might hope that an explicit description of the E_{1}-terms might provide a modest first step toward the monodromy-weight conjecture for these Shimura varieties. The methods of this paper allow us to completely describe the semisimplification (gr_{i}^{W})^{ss} of each graded piece gr_{i}^{W} (RΨ), in the category P (Sh ×_{s} η, \overline{ℚ}_{ℓ}). This can be used to describe the E_{1}-terms, if one forgets the Galois action. The following summarizes our results in this direction. (We will write m(w, i) for m(RΨ, w, i) here.)

Proposition 8.1 In the category P (Sh ×_{s} η, \overline{ℚ}_{ℓ}), we have the identity

(gr_{i}^{W})^{ss} = ⨁_{{w∈Adm(μ) \atop ℓ(w)+2j=i}} IC_{Sh,w}(−j)^{⊕m(w,j)}.

Here IC_{Sh,w} is the intersection complex of the closure of the Kottwitz-Rapoport stratum Sh_{w} in Sh_{\overline{𝔽}_{p}}. Consequently, if Sh is proper over O_{E}, we deduce (forgetting the Galois actions) isomorphisms of \overline{ℚ}_{ℓ}-spaces

W E_{1}^{pq} = ⨁_{{w∈Adm(μ) \atop ℓ(w)+2j=−p}} IH^{p+q+ℓ(w)}(\overline{Sh}_{w}, \overline{ℚ}_{ℓ})^{⊕m(w,j)}.

These results illustrate the fundamental importance of the Kottwitz-Rapoport stratification on Sh𝔽̄_{p}. That stratification is indexed by the familiar set Adm(μ), but the strata themselves are still rather mysterious. In future work we will examine these strata more closely.

9 Examples

In this section we present a few examples for multiplicities of nearby cycles sheaves RΨ = RΨ(IC(Q_{μ})[ℓ(t_{μ})]). Most of the examples are for G = GL_{n}, but there is one example for each of GSp_{4} and GSp_{6}, and also one for G of type G_{2}.

9.1 The case of a divisor with normal crossings

Proposition 9.1 In the Drinfeld case, where GL_{n} and μ = (1, 0, …, 0), we have m(x, i) = 1 for all x ∈ Adm(μ), 0 ≤ i ≤ ℓ(t_{μ}) − ℓ(x), and m(x, i) = 0 otherwise.

Proof. This is a combinatorial exercise using the following facts. In this case all Kazhdan-Lusztig polynomials are 1 (since the special fiber is a union of smooth divisors with normal crossings) and Tr(Fr_{q}, RΨ_{x}) = ε_{μ}(1 − q)^{ℓ(t_{μ})−ℓ(x)} (cf. [H3], Prop. 5.2). □

9.2 GL_{2}

Proposition 9.2 In the case of GL_{2} and an arbitrary coweight μ, all multiplicities m(x, i) = 1 for x ∈ Adm(μ) and 0 ≤ i ≤ ℓ(t_{μ}) − ℓ(x), and m(x, i) = 0 otherwise.

Proof. Argue as above, using the explicit formula for Tr(Fr_{q}, RΨ_{x}) given in [H1], Prop. 10.3. □

9.3 Other examples

Further examples can be computed in the following way: the Kottwitz conjecture (see section 2.7) describes the semi-simple trace of Frobenius on RΨ, and since the inertia action is unipotent, this is the same as the usual trace. So the function of RΨ is known, and thus the multiplicities can be computed by applying the base change from the T_{w}-basis to the C′′_{w}-basis in the Hecke algebra. In order to carry this out, though, Bernstein functions and lots of Kazhdan-Lusztig polynomials have to be computed, so it is almost impossible to do this manually. The following examples were computed with the help of a computer program.

Because in all the interesting examples the number of alcoves is quite large, we did not print out the multiplicities for each single alcove, but we subsumed the information for several alcoves into one line.

Besides the length l and the number of alcoves that have certain multiplicities, we also put the ’Bruhat configuration’ in the tables, i.e. the number of alcoves of length l + 1, l + 2, … which are greater than the given alcove with respect to the Bruhat order; cf. the remarks below.

The multiplicities listed below are always the numbers m(w, i) where i is understood to range from 0 to ℓ(t_{μ}) − ℓ(w) (the multiplicities for i not in this range always happen to vanish, cf. section 9.4 for some discussion of this point).

9.3.1 G = GL_{4}, μ = (1, 1, 0, 0)

Number of admissible alcoves: 33

Length #Alcoves Multiplicities Bruhat configuration
l=0  1  1,1,2,1,1  4, 10, 12, 6
l=1  4  1,2,2,1  5, 8, 5
l=2  8  1,1,1  3, 3
2  1,2,1  4, 4
l=3  12  1,1  2
l=4  6  1  -

This is the simplest non-trivial case. We see already that the multiplicities for w are not determined by the length of w.

9.3 Other examples 43

9.3.2 G = GL_{5}, μ = (1, 1, 0, 0, 0) Number of admissible alcoves: 131

Length #Alcoves  Multiplicities  Bruhat configuration
l=0  1  1, 1, 2, 2, 2, 1, 1 5, 15, 30, 40, 30, 10
l=1  5  1, 2, 3, 3, 2, 1 6, 17, 28, 24, 9
l=2  5  1, 1, 2, 1, 1  4, 10, 12, 6
5  1, 2, 2, 2, 1  6, 14, 15, 7
5  1, 3, 4, 3, 1  7, 17, 18, 8
l=3  5  1, 1, 1, 1  4, 6, 4
20  1, 2, 2, 1  5, 8, 5
5  1, 2, 2, 1  5, 9, 6
l=4  30  1, 1, 1  3, 3
10  1, 2, 1  4, 4
l=5  30  1, 1  2
l=6  10  1  -

9.3.3 G = GL_{6}, μ = (1, 1, 0, 0, 0, 0) Number of admissible alcoves: 473

Length #Alcoves  Multiplicities  Bruhat configuration
l=0  1  1, 1, 2, 2, 3, 2, 2, 1, 1 6, 21, 50, 90, 120, 110, 60, 15
l=1  6  1, 2, 3, 4, 4, 3, 2, 1 7, 24, 55, 86, 88, 52, 14
l=2  6  1, 1, 2, 2, 2, 1, 1  5, 15, 30, 40, 30, 10
6  1, 2, 3, 3, 3, 2, 1  7, 24, 48, 58, 39, 12
9  1, 3, 5, 6, 5, 3, 1  8, 28, 56, 67, 44, 13
l=3  30  1, 2, 3, 3, 2, 1  6, 17, 28, 24, 9
6  1, 2, 2, 2, 2, 1  7, 20, 30, 24, 9
12  1, 3, 4, 4, 3, 1  8, 25, 39, 31, 11
2  1, 4, 7, 7, 4, 1  9, 30, 47, 36, 12
l=4  15  1, 1, 2, 1, 1  4, 10, 12, 6
6  1, 1, 1, 1, 1  5, 10, 10, 5
30  1, 2, 2, 2, 1  6, 14, 15, 7
6  1, 2, 2, 2, 1  6, 14, 16, 8
3  1, 2, 3, 2, 1  6, 15, 18, 9
30  1, 3, 4, 3, 1  7, 17, 18, 8
l=5  30  1, 1, 1, 1  4, 6, 4
60  1, 2, 2, 1  5, 8, 5
30  1, 2, 2, 1  5, 9, 6
l=6  80  1,1,1  3, 3
30  1,2,1  4, 4
l=7  60  1,1  2
l=8  15  1  -

44 9 EXAMPLES

9.3.4 G = GL_{3}, μ = (2, 2, 0) Number of admissible alcoves: 19

Length #Alcoves Multiplicities Bruhat configuration
l=0  1  1, 1, 2, 1, 1 3, 6, 6, 3
l=1  3  1,2,2,1  4, 5, 3
l=2  3  1,1,1  2, 2
3  1,1,1  3, 3
l=3  6  1,1  2
l=4  3  1  -

9.3.5 G = GL_{3}, μ = (3, 1, 0) Number of admissible alcoves: 49

Length #Alcoves  Multiplicities  Bruhat configuration
l=0  1  1, 2, 3, 3, 3, 2, 1 3, 6, 9, 12, 12, 6
l=1  3  1, 3, 5, 5, 3, 1 4, 8, 12, 12, 6
l=2  3  1, 2, 3, 2, 1  4, 9, 11, 6
3  1, 3, 4, 3, 1  4, 9, 11, 6
l=3  3  1,2,2,1  4, 6, 4
3  1,2,2,1  4, 8, 6
3  1,4,4,1  6, 10, 6
l=4  3  1,1,1  2, 2
9  1,2,1  4, 4
l=5  12  1,1  2
l=6  6  1  -

This example shows that the Bruhat configuration numbers given in the table do not determine the multiplicities (look at the elements of length 2). On the other hand, in this example it is still quite easy to check that the Bruhat graphs for the two types of elements are different. 9.3.6 G = GL_{4}, μ = (2, 0, 0, 0) Number of admissible alcoves: 65

Length #Alcoves  Multiplicities  Bruhat configuration
l=0  1  1, 1, 2, 2, 2, 1, 1 4, 10, 16, 18, 12, 4
l=1  4  1, 2, 3, 3, 2, 1 5, 11, 15, 11, 4
l=2  4  1, 1, 2, 1, 1  3, 6, 6, 3
4  1, 2, 2, 2, 1  5, 10, 9, 4
2  1, 3, 4, 3, 1  6, 12, 10, 4
l=3  12  1,2,2,1  4, 5, 3
4  1,1,1,1  4, 6, 4
l=4  6  1,1,1  2, 2
12  1,1,1  3, 3
l=5  12  1,1  2
l=6  4  1  -

9.3 Other examples 45

9.3.7 G = GL_{4}, μ = (2, 1, 0, 0) Number of admissible alcoves: 143

Length #Alcoves  Multiplicities  Bruhat configuration
l=0  1  1, 2, 3, 4, 4, 3, 2, 1 4, 10, 20, 30, 36, 30, 12
l=1  4  1, 3, 5, 6, 5, 3, 1 5, 14, 25, 33, 29, 12
l=2  4  1, 3, 4, 4, 3, 1  5, 12, 20, 21, 10
4  1, 2, 3, 3, 2, 1  5, 14, 25, 26, 12
2  1, 6, 11, 11, 6, 1 8, 20, 30, 28, 12
l=3  4  1,2,2,2,1  3, 6, 9, 6
4  1,2,3,2,1  5, 13, 17, 9
12  1,3,4,3,1  6, 15, 19, 10
l=4  12  1,2,2,1  4, 8, 6
8  1,2,2,1  5, 9, 6
8  1,3,3,1  6, 11, 7
2  1,3,3,1  6, 12, 8
l=5  8  1,1,1  3, 3
28  1,2,1  4, 4
l=6  30  1,1  2
l=7  12  1  -

9.3.8 G = GSp_{4}, μ = (1, 1, 0, 0) Number of admissible alcoves: 13

Length #Alcoves Multiplicities Bruhat configuration
l=0  1  1, 1, 1, 1 3, 5, 4
l=1  2  1, 1, 1  3, 3
1  1, 2, 1  4, 4
l=2  5  1,1  2
l=3  4  1  -

46 9 EXAMPLES

9.3.9 G = GSp_{6}, μ = (1, 1, 1, 0, 0, 0) Number of admissible alcoves: 79

Length #Alcoves Multiplicities Bruhat configuration
l=0  1  1,1,1,2,1,1,1 4, 9, 17, 22, 18, 8
l=1  2  1,1,2,2,1,1 4, 10, 16, 15, 7
2  1,2,3,3,2,1 5, 13, 19, 17, 8
l=2  1  1,1,2,1,1  4, 10, 12, 6
4  1,2,2,2,1  5, 10, 11, 6
2  1,2,2,2,1  5, 11, 14, 8
2  1,3,4,3,1  6, 13, 14, 7
l=3  6  1,1,1,1  3, 5, 4
2  1,1,1,1  4, 6, 4
4  1,2,2,1  5, 8, 5
4  1,2,2,1  5, 9, 6
1  1,3,3,1  6, 12, 8
l=4  14  1,1,1  3, 3
8  1,2,1  4, 4
l=5  18  1,1  2
l=6  8  1  -

9.3.10 G of type G_{2}, μ = (2, 1, 0) Number of admissible alcoves: 41

Length #Alcoves  Multiplicities  Bruhat configuration
l=0  1  1, 1, 1, 1, 1, 1, 1 3, 5, 7, 9, 10, 6
l=1  1  1, 1, 1, 1, 1, 1 3, 5, 7, 9, 6
1  1, 1, 1, 1, 1, 1 3, 6, 9, 10, 6
1  1, 2, 2, 2, 2, 1 4, 7, 9, 10, 6
l=2  2  1,1,1,1,1  3, 5, 7, 5
1  1,2,2,2,1  4, 7, 9, 6
2  1,2,2,2,1  4, 8, 10, 6
l=3  3  1,1,1,1  3, 5, 4
2  1,2,2,1  4, 7, 5
1  1,2,2,1  4, 8, 6
1  1,2,2,1  5, 9, 6
l=4  4  1,1,1  3, 3
5  1,2,1  4, 4
l=5  10  1,1  2
l=6  6  1  -

9.4 Numerical Observations and Conjectures

Looking at the tables, one can make several empirical observations: (A) We have m(w, i) ∈ [0, ℓ(μ) − ℓ(w)], or equivalently, the function

Tr(Fr_{q}, RΨ_{w}) is a polynomial in q of degree ≤ ℓ(μ) − ℓ(w); (B) The sequence m(w, 0), …, m(w, ℓ(μ) − ℓ(w)) is palindromic; it increases

to the middle, then decreases again.

(C) We have m(w, 0) = m(w, ℓ(μ) − ℓ(w)) = 1.

From some more detailed information not contained in the tables, we also made the following striking observation:

• The multiplicity polynomial m(w) = ∑_{i} m(w, i)q^{i} is determined by the isomorphism type of the Bruhat graph of the set {x ∈ Adm(μ) | x ≥ w} (although not by the Bruhat configuration numbers alone). It is remarkable that the multiplicity polynomials seem to be determined by data that is independent of the underlying root system.

Using the explicit formula for Tr(Fr_{q}, RΨ_{w}) in the minuscule case, proved in [H3],[HP], we can prove this last observation in that case. Based on our data we conjecture it holds in complete generality. We offer the following explanations for (A)-(C).

Proof of (A) for GL_{n}, or for μ minuscule.

From the definition, we get a recursive procedure to compute the multiplicities:

ε_{w} ∑_{i} m(w, i)q^{i} = Tr(Fr_{q}, RΨ_{w}) − ∑_{{x∈Adm(μ) \atop x>w}} ε_{x}m(x)P_{w,x}.

By downward induction on ℓ(w), we deduce that Tr(Fr_{q}, RΨ_{w}) is a polynomial of degree ≤ ℓ(μ) − ℓ(w) for all w ∈ Adm(μ) if and only if m(w) is a polynomial of degree ≤ ℓ(μ) − ℓ(w), for all such w.

Now suppose μ is either a minuscule coweight, or an arbitrary coweight for the group GL_{n}. In [H3] and [HP] it is proved that for any coweight ν ∈ Ω(μ), the function Θ_{ν} has a minimal expression. Therefore by Proposition 7.6, we see that

Θ_{ν}(w) = q_{w}^{−1/2}R(Q),

where R(Q) ∈ ℕ[Q], and deg_{Q}R(Q) ≤ ℓ(t_{ν}) − ℓ(w). From this it is easy to see (via the Kottwitz conjecture), that Tr(Fr_{q}, RΨ_{w}) is a polynomial in q of degree ≤ ℓ(μ) − ℓ(w).

We remark that the conclusion also holds when every dominant coweight λ ≤ μ is a sum of dominant minuscule coweights.

Some time after this article had been submitted, we found a proof which works without using minimal expressions and hence works without the above restriction on G or μ [GH]. Furthermore, it follows from recent results of Ram [Ra] that in fact minimal expressions always exist; hence the proof above is applicable to the general case.

Proof of (B).

Recall that DRΨ = RΨ(ℓ(μ)). Using Lemma 4.5, we deduce immediately that the multiplicity function m(RΨ, w) := ∑_{i} m(RΨ, w, i)q^{i} is palindromic, i.e., \overline{m(RΨ, w)} = q_{μ}^{−1}q_{w}m(RΨ, w).

The multiplicities being palindromic is also a consequence of the conjecture that the monodromy and the weight filtration on nearby cycles agree. In the function field case this is known to be true (cf. [BB] §5); since the multiplicities in the unequal characteristic case and in the function field case coincide, we get the same consequences in the unequal characteristic case. In particular this proves also that the multiplicities increase to the middle, and then decrease again.

Proof of (C) in the minuscule case.

It is easy to see that, for minuscule μ, m(w, l(μ) − l(w)) = 1 for all w because then Tr(Fr_{q}, RΨ_{w}) is essentially an R-polynomial ([H3], [HP]), and thus the coefficient of q^{l(μ)−l(w)} in Tr(Fr, RΨ_{w}) is ε_{w}.

This also implies m(w, 0) = 1 for all w, since the multiplicities are palindromic.

If μ is minuscule, the identity m(x, 0) = 1 for all x ∈ Adm(μ) is equivalent to the following combinatorial identity for such x:

∑_{{w∈Adm(μ) \atop w≥x}} ε_{w} = ε_{μ}.

This has certain geometric consequences. For example, it shows that any codimension 1 Iwahori-orbit in a local model is contained in at least two irreducible components. It follows from this and properties of the Bruhat order that the smooth locus of a local model is precisely the union of the extreme strata. Consequently, we get the analogous results for certain Shimura varieties. (See [GN] for related results, proved by a different method.)

Proposition 9.3 Let Sh be a Shimura variety as in section 8. Then the smooth locus of the special fiber of Sh is precisely the union of the Kottwitz-Rapoport strata indexed by the translation elements in the set Adm(μ). Here μ is the minuscule coweight in the Shimura datum for Sh.

10 Appendix

Part I: Nearby cycles preserve mixedness

Due to fundamental results of Deligne [Weil2], it is known that nearby cycles preserve mixedness in the function-field setting (i.e., when working over a trait which is the Henselization at a point of a smooth curve over a finite field). The aim of this appendix is to prove an analogous result in the setting of unequal characteristic traits. This result applies to nearby cycles on local models and integral models for Shimura varieties over p-adic number rings, ensuring that they possess weight filtrations, as postulated in the introduction.

Let (S, s, η) denote a Henselian trait (i.e, the spectrum of a complete discrete valuation ring). We assume that the residue field k(s) is finite. Let k(η̄) be a separable closure of k(η), and let S̄ denote the normalization of S in η̄, with closed point s̄. Let X → S be a separated, finite-type scheme over S.

We will use the notion of S-variety (cf. [deJ2], 2.15). A separated finite-type S-scheme X will be called an S-variety if X → S is flat and X is integral.

For A ∈ D_{c}^{b}(X_{η}, ℚ̄_{ℓ}), define RΨ^{X}(A) = ī^{∗}Rj̄_{∗}Aη̄, an object in D_{c}^{b}(X ×_{s} η, ℚ̄_{ℓ}).

Let Λ_{η} denote the constant ℓ-adic sheaf ℚ̄_{ℓ} on X_{η} (or some λ-adic constant sheaf E_{λ}, for λ|ℓ) We will prove the following theorem.

Theorem 10.1 Suppose that X_{η} is smooth. Then the complex RΨ^{X}(Λ_{η}) is mixed.

The proof of the theorem consists of several steps which eventually reduce it to the calculations of Rapoport-Zink ([RZ1], [I]) for the case of a proper strictly semi-stable S-variety (for the precise definition of strictly semi-stable, see [deJ2], 2.16). We will use de Jong’s alterations [deJ2] to reduce the general case to that case. Roughly, if X/S is a proper S-variety, one takes a strictly semi-stable alteration π : X′ → X, and then applies the push-forward functor π_{∗} to appropriate nearby cycles on the alteration. Thanks to Rapoport-Zink and properties of the push-forward functor, the resulting complex is mixed. One proves using the trace map that it contains the original nearby cycles as a subobject, thus essentially completing the proof of the theorem in that case. (An extra difficulty arises because in the end we have to descend from a finite extension k(s′) back down to k(s); see Corollary 10.5.) It is convenient to work with perverse sheaves throughout this process. In particular, we will work with the perverse push-forward ^{p}π_{∗}.

10.1 Alterations

The following theorem is a direct consequence of [deJ2], Theorem 6.5.

Theorem 10.2 (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.}

10.2 Proof for X′/S′

Let X′, S′ etc. be as in Theorem 10.2. The special fiber \overline{X′}_{s′} is globally a union of reduced divisors with normal crossings and the calculations of Rapoport-Zink are valid (cf. [I], Th. 3.2). The sheaves R^{q}Ψ\overline{X′}(Λ_{η′}) are explicitly computed in loc. cit, and are easily seen to be mixed. Thus Theorem 10.1 holds in this case.

This implies the theorem for X′/S′. By Theorem 10.2 the morphism j embeds X′ as an open subscheme in \overline{X′}. Then the formula

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

shows that the right hand side is mixed.

10.3 Proof for X_{η} smooth and geometrically integral

Assume that X/S is finite-type and separated, with X_{η} smooth and geometrically integral. The nearby cycles are supported on the scheme-theoretic closure of X_{η} in X_{s}. We may therefore assume X/S is flat by replacing X with the scheme-theoretic closure of X_{η} in X (which is flat with the same generic fiber since X_{η} is reduced). Thus, we may assume X/S is an S-variety whose generic fiber is smooth and geometrically integral.

Let S′ and X′ be as in Theorem 10.2.

Let us denote by π : X′ → X_{S′} the resulting alteration of S′-varieties. Let us also denote by π : X′_{η′} → X_{η′} the morphism on the generic fibers. This is an alteration, hence by definition is generically finite, say of degree n. Since X_{η′} is integral, π is also generically flat. More precisely, there exists a non-empty open subset U ⊂ X_{η′} such that π : π^{−1}(U) → U is flat and finite.

10.3 Proof for X_{η} smooth and geometrically integral 51

Let us denote ^{p}π^{∗} := ^{p}H^{0}π^{∗} (resp. ^{p}π_{∗} := ^{p}H^{0}π_{∗}), the perverse versions of the pull-back and push-forward functors via π. Since π is proper, we have π_{∗} = π_{!} and so ^{p}π_{∗} = ^{p}π_{!}.

Let Λ_{η′}^{X} denote the constant sheaf on X_{η′}, and let Λ_{η′}^{X′} denote the constant sheaf on X′_{η′}. For d = dim(X_{η}), let A^{X} = Λ_{η′}^{X}[d](\frac{d}{2}), and A^{X′} = Λ_{η′}^{X′}[d](\frac{d}{2}); the smoothness of X′_{η′} (resp. X_{η′}) ensures that A^{X′} (resp. A^{X}) is perverse of weight zero. We will need the following lemma.

Lemma 10.3 The perverse sheaf A^{X} is a subobject of ^{p}π_{∗}(A^{X′}), in the category P (X_{η′}, \overline{ℚ}_{ℓ}).

Proof. The adjunction morphism A^{X} → π_{∗}π^{∗}A^{X} and the perversity of A^{X} yield a morphism of perverse sheaves A^{X} → ^{p}π_{∗}A^{X′}. Since A^{X} is simple, it is sufficient to prove this morphism is not the zero morphism. This can be detected by showing that for some dense open U ⊂ X_{η′}, the restriction of the morphism to U is nonzero. Now choose U over which π is flat and finite, and use the trace map

Tr : (π_{!}π^{∗}(Λ_{η′}^{X}))|_{U} → (Λ_{η′}^{X})|_{U},

which composed with the canonical adjunction map Λ_{η′}^{X} → π_{∗}π^{∗}(Λ_{η′}^{X}) = π_{!}π^{∗}(Λ_{η′}^{X}) is just multiplication by the degree n. For the definition and basic properties of this trace map, see [SGA 4, Exp. XVII, p. 553-554]. □

Now let r : X_{S′} → X denote the projection morphism. Since r arises from a finite “change of trait” morphism, the invariance of nearby cycles under change of trait ([SGA 4 1/2, Thm. finitude 3.7]) implies that we have a canonical isomorphism r^{∗}RΨ^{X}(Λ_{η}[d](\frac{d}{2})) = RΨ^{X_{S′}} (A^{X}) in the category D_{c}^{b}(X ×_{s} η′, ℚ̄_{ℓ}). Here the meaning of this last category and of the left hand side of the equality is the following. The nearby cycles RΨ^{X}(Λ_{η}[d](\frac{d}{2})) live on the geometric special fiber Xs̄ = Xs̄′, and naturally carry a compatible action of Gal(η̄/η). The sheaf r^{∗}RΨ^{X}(Λ_{η}[d](\frac{d}{2})) is just that same sheaf together with the compatible action of Gal(η̄′/η′) via the canonical restriction map Gal(η̄′/η′) → Gal(η̄/η). By Lemma 10.3, this is a subobject (in the category P (X ×_{s′} η′, \overline{ℚ}_{ℓ})) of

RΨ^{X_{S′}} (^{p}π_{∗}(A^{X′})) = ^{p}π_{∗}RΨ^{X′}(A^{X′}).

(The last equality follows from the compatibility of nearby cycles and proper push-forwards together with the fact that RΨ commutes with the perverse truncation functors ^{p}τ_{≤0} and ^{p}τ_{≥0}, hence with the functor ^{p}H^{0}. The compatibility of RΨ and the perverse truncation functors follows easily from the fact that RΨ preserves ^{p}D_{c}^{≤0} and commutes with Verdier duality.)

We have used here the basic fact that the functor RΨ, being t-exact, induces an exact functor on categories of perverse sheaves.

We can now complete the argument. By the case of X′/S′ treated above, the complex RΨ^{X′}(A^{X′}) is mixed. Thus by [BBD] §5, ^{p}π_{∗}RΨ^{X′}(A^{X′}) is mixed. Its subobject r^{∗}RΨ^{X}(Λ_{η}[d](\frac{d}{2})) is mixed too. By the lemma and corollary in the following section on descent, we deduce finally that RΨ^{X}(Λ_{η}) is also mixed. This proves Theorem 10.1 in the case of finite-type separated S-schemes X where X_{η} is smooth and geometrically integral.

10.4 Proof for X_{η} smooth

We consider finally the general case, where X_{η} is only assumed smooth. Since Xη̄ is smooth, its connected components are integral. If η′/η is a finite extension over which all the connected components are defined, then it follows that each connected component of X_{η′} is smooth and geometrically integral. Let S′ denote the normalization of S in η′; then S′ is a trait, finite over S. As above let r : X_{S′} → X denote the projection. For each connected component X_{η′}^{(α)}, α = 1, 2, …, m, let X^{(α)} denote its scheme-theoretic closure in X_{S′}. We have closed immersions of S′-schemes i_{α} : X^{(α)} ↪ X_{S′}. Each X^{(α)} is a separated finite-type S′-scheme whose generic fiber is smooth and geometrically integral; hence the results of the previous section obtain for it.

It is clear that

RΨ^{X_{S′}} (Λ_{η′}) = ⨁_{α} i_{α,∗}RΨ^{X^{(α)}}(Λ_{η′});

by the previous section the right hand side is mixed, hence so is the left hand side. Further, as before, r^{∗}RΨ^{X}(Λ_{η}) = RΨ^{X_{S′}} (Λ_{η′}) is just the complex RΨ^{X}(Λ_{η}) endowed with its compatible action of Gal(η̄′/η′) via Gal(η̄′/η′) → Gal(η̄/η). Now Corollary 10.5 finally proves that RΨ^{X}(Λ_{η}) is mixed, as desired.

10.5 Descent step

The following results are well-known (cf. [KW], p.14); we include them for the reader’s convenience.

Lemma 10.4 Let r : S′ → S denote a finite surjective morphism of traits with finite residue fields. Let X/S be finite-type, and by abuse denote also by r the morphism X_{s′} → X_{s} on special fibers deduced from r ×_{S} id_{X} : X_{S′} → X.

If G is a smooth étale Weil sheaf on X_{s} such that r^{∗}G is a mixed sheaf on X′_{s′} with weights ≤ w, then G is also mixed with weights ≤ w.

We suppose k(s) has q elements. As noted above in section 10.3, because Xs̄ = Xs̄′, we can regard G as a sheaf on Xs̄ endowed with a compatible action of a geometric Frobenius Frob_{q} ∈ Gal(η̄/η), and we can regard r^{∗}G as this same sheaf endowed with a compatible action of a geometric Frobenius in Gal(η̄′/η′) which maps to Frob_{q}^{n} = Frob_{q^{n}} under the restriction map Gal(η̄′/η′) → Gal(η̄/η), where n = [k(s′) : k(s)].

Proof. Let X_{0} = X_{s} = s ×_{S} X. Note that the “change of trait” morphism r above is the morphism

s′ ×_{S′} X′ → s ×_{S} X

which arises via the canonical identifications s′×_{S′}(S′×_{S}X) = s′×_{S}X = s′×_{s}X_{s} from the projection

r : X_{0} ×_{κ} κ′ → X_{0},

where X_{0} = X_{s}, κ = 𝔽_{q}, and κ′ = 𝔽_{q^{n}}.

The sheaf r^{∗}G is by hypothesis a smooth mixed Weil sheaf on X_{0} ×_{κ} κ′. We want to prove that G is a mixed Weil sheaf on X_{0}.

Since r^{∗} is exact, we can assume without loss of generality that G is simple. We will prove that in this case G is pure.

Since r^{∗}G is mixed, there is a non-zero smooth Weil sheaf F on X_{0} ×_{κ} κ′ which is a simple (thus pure) Weil subsheaf of r^{∗}G.

Let now F denote the Frobenius automorphism Fr_{q} for X_{0}, so that F^{n} is the Frobenius automorphism for X_{0} ×_{κ} κ′. We are given an isomorphism (F^{n})^{∗}F ˜ ⟶F. Using this and the fact that the category of Weil sheaves is closed under the formation of kernels and cokernels, one can show that

∑_{i=0}^{n−1}(F^{i})^{∗}F

is a nonzero Weil subsheaf of G. Since the latter is simple, we have in fact

⨁_{i∈I}(F^{i})^{∗}F = G,

where I ⊂ {0, 1, …, n − 1} is a nonempty subset. (The sum can be made a direct sum by omitting certain i’s.)

Now fix any isomorphism τ : ℚ̄_{ℓ} ˜ ⟶ℂ. Note that the summands above are pure Weil sheaves on X_{0} ×_{κ} κ′, whose τ-weights are all identical. Suppose each is τpure of weight w. For any geometric point x̄ over a closed point x ∈ X_{0}, we set F_{x} = F^{[κ(x):κ]}; then for any eigenvalue α of F_{x}^{n} on an irreducible summand of Gx̄, we have |τ(α)| = q^{nd(x)w/2}. A similar statement thus holds for F_{x} replacing F_{x}^{n} and q^{wd(x)/2} replacing q^{wd(x)n/2}. This holds for every choice of x, and w is independent of that choice. It follows that G is pure of weight w. □

Corollary 10.5 Let r, X be as above. Let G ∈ D_{c}^{b,W eil}(X_{s}, ℚ̄_{ℓ}). If r^{∗}G is a mixed complex in D_{c}^{b,W eil}(X_{s′}, ℚ̄_{ℓ}), then G is also mixed.

Proof. By passing to cohomology sheaves, we can reduce to the case where G and r^{∗}G are constructible Weil sheaves on X_{s} and X′_{s′}, respectively. By Noetherian induction and gluing of constructible sheaves, we can reduce further to the case where G and thus r^{∗}G is a smooth Weil sheaf, where the result was proved in the foregoing lemma. □

10.6 A variant

The following variant applies to the nearby cycles appearing in the local models and deformations of affine flag varieties treated in this paper. It does not apply to Shimura varieties: while the generic fibers of these are smooth, they are usually not geometrically integral.

Theorem 10.6 Suppose X_{η} is geometrically integral, with intersection complex 0IC(X_{η}). Then RΨ^{X}(^{0}IC(X_{η})) is mixed.

Here, ^{0}IC(X_{η}) denotes the intersection complex of X_{η}, suitably Tate-twisted and shifted to make it a pure self-dual perverse sheaf of weight zero. This normalization is convenient but of course not necessary for the theorem.

Proof. The proof works the same way as before. First, replacing X with the scheme-theoretic closure of X_{η}, we may assume X/S is flat, hence integral, i.e., we may assume X is an S-variety with geometrically integral generic fiber.

Now consider the alterations π : X′ → X_{S′} and π : X′_{η′} → X_{η′} associated to the diagram in Theorem 10.2. Let r : X_{S′} → X denote the projection morphism. Arguing almost as before (see the Lemma below) we can use the trace map to prove that ^{0}IC(X_{η′}) = ^{0}IC(X_{η})_{η′} is a subquotient of ^{p}π_{∗}A^{X′}. By invariance of nearby cycles under change of traits, r^{∗}RΨ^{X}(^{0}IC(X_{η})) = RΨ^{X_{S′}} (^{0}IC(X_{η′})), which is a subquotient of RΨ^{X_{S′}} (^{p}π_{∗}A^{X′}) = ^{p}π_{∗}RΨ^{X′}(A^{X′}). The latter is mixed by section 10.2. Applying descent as in section 10.5, the result follows. □

Lemma 10.7 The perverse sheaf ^{0}IC(X_{η′}) is a subquotient of ^{p}π_{∗}(A^{X′}), in the category P (X_{η′}, \overline{ℚ}_{ℓ}).

Proof. Choose the open set U ⊂ X_{η′} as in Lemma 10.3. Let j′ : U′ := π^{−1}(U) → X′_{η′} be the open immersion. Note that IC := ^{0}IC(X_{η′}) is just j_{!∗}A^{U}, and A^{X′} = j′_{!∗}A^{U′}, where A^{U}, resp. A^{U′} has the obvious meaning.

By using adjunction maps and the morphism of functors j_{!∗} → j_{∗}, we get morphisms

IC → ^{p}π_{∗}π^{∗}j_{!∗}A^{U} → ^{p}π_{∗}π^{∗}j_{∗}A^{U} → ^{p}π_{∗}j′_{∗}π^{∗}A^{U} = ^{p}π_{∗}j′_{∗}A^{U′},

and

pπ_{∗}A^{X′} = ^{p}π_{∗}j′_{!∗}A^{U′} → ^{p}π_{∗}j′_{∗}A^{U′}.

Now restricting to U with j^{∗} and using the trace map as before shows that the first map is non-zero, hence a monomorphism. Also, restricting shows that the image of IC under the first map lies in the image of the second map (if not, then IC embeds into the cokernel of the second map, which is ruled out after restricting both to U). Thus, IC is a subquotient of ^{p}π_{∗}A^{X′}. □

Part II: A bound on the weights of nearby cycles

The method of proof of Theorem 10.6 gives an estimate on the weights:

w(RΨ^{X}(^{0}IC(X_{η}))) ≤ dim(X_{η}).

(One is reduced to the proper strictly semi-stable case, where it is easy).

Now let RΨ = RΨ(IC(Q̄_{μ})[ℓ(t_{μ})]), as usual.

The fact (proved in section 7.2) that for μ either minuscule, or a coweight for GL_{n}, the function Tr(Fr_{q}, RΨ_{x}) is polynomial in q of degree ≤ ℓ(μ) − ℓ(x), is equivalent to the following much stronger bound on the weights of ^{0}RΨ := RΨ(\frac{ℓ(μ)}{2}): for each x ∈ Adm(μ), and each closed point z ∈ B_{x}, the weights of the complex ^{0}RΨ_{z} are ≤ ℓ(μ)−ℓ(x). In all the examples we computed, it is true that the trace of Frobenius is always a polynomial in q with this degree. In all the cases we have proved this (thus far), it follows from the existence of minimal expressions for the functions Θ_{λ} in the Iwahori-Hecke algebra (cf. section 9.4). It would be interesting to find a direct geometric argument to prove this sharper bound on the weights in all cases.^{4}

The following gives a sufficient condition for the sharp bound to hold. In this statement, we assume there is the action of a geometrically connected smooth group scheme G/S on X/S, and we say a Gs̄-equivariant Weil perverse sheaf K on Xs̄ satisfies the sharp bound if for each locally closed immersion i : O → Xs̄ of a Gs̄-orbit O, we have w(i^{∗}K) ≤ dim(Xs̄) − dim(O).

Proposition 10.8 Let X/S be an S-variety such that X_{η} is geometrically integral. Suppose that X/S carries an action of a geometrically connected smooth group scheme G/S. Suppose that there exists a trait S′, finite over S, and an S-alteration X′ → X with compactification \overline{X′} as in Theorem 10.2, each carrying an action of G_{S′}, such that the inclusion j : X′ ↪ \overline{X′} and the S′-alteration π : X′ → X_{S′} are G_{S′}-equivariant. Then RΨ^{X}(^{0}IC(X_{η})) satisfies the sharp bound on its weights.

Proof. This is similar to the proof of Theorem 10.6. The sharp bound holds for the strictly semi-stable case, and thus also for RΨ^{X′}(A^{X′}). It is enough to prove that the sharp bound holds for ^{p}π_{!}RΨ^{X′}(A^{X′}). So, let O ⊂ X_{s′} denote a G_{s′}- orbit of dimension m, and suppose z ∈ O is a closed point. Using equivariance, one can show that any G_{s′}-orbit contained in π^{−1}(O) dominates O, hence has dimension ≥ m. Thus, the weights of π_{!}RΨ^{X′}(A^{X′})_{z} = RΓ_{c}(π^{−1}(z), RΨ^{X′}(A^{X′})) are ≤ dim(X′_{s̄′}) − m, and we are done. □

There are B-equivariant semi-stable resolutions for the local models M^{loc} = M_{μ}, in the cases GSp_{4}, μ = (1, 1, 0, 0) [deJ1], and GSp_{6}, μ = (1, 1, 1, 0, 0, 0) [Ge]. Together with the above considerations, we get the following corollary.

4In the meantime, we have found such a proof, see [GH].

Corollary 10.9 Let μ be the minuscule coweight (1, 1, 0, 0) for GSp_{4} or the minuscule coweight (1, 1, 1, 0, 0, 0) for GSp_{6}. Then the sharp bound holds for the weights of ^{0}RΨ. In particular, in each of these cases the multiplicity functions

∑_{i} m(RΨ, w, i)q^{i}

are polynomials of degree at most ℓ(μ) − ℓ(w), for every w ∈ Adm(μ).

This Corollary also follows from results already announced in section 9, but the approach given here is perhaps more conceptual.

Acknowledgements This work was initiated during our visits to the Institute for Advanced Study in 2000-2001. We are grateful to the IAS for its hospitality and financial support. The research of the second author was also partially supported by NSF grant DMS-0303605.

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Ulrich Görtz: Mathematisches Institut der Universität Bonn, Beringstr. 6, 53115 Bonn, Germany. Email: ugoertz@math.uni-bonn.de 60 REFERENCES Thomas Haines: Mathematics Department, University of Maryland, College Park, MD 20742-4015, USA. Email: tjh@math.umd.edu