Nearby cycles for local models of some Shimura varieties

T. Haines, B.C. Ngô

Abstract Kottwitz conjectured a formula for the (semi-simple) trace of Frobenius on the nearby cycles for the local model of a Shimura variety with Iwahori-type level structure. In this paper, we prove his conjecture in the linear and symplectic cases by adapting an argument of Gaitsgory, who proved an analogous theorem in the equal characteristic case.

AMS Classification : 11G18, 14G35, 20C08, 14M15

1 Introduction

For certain classical groups G and certain minuscule coweights μ of G, M. Rapoport and Th. Zink have constructed a projective scheme M(G, μ) over ℤ_{p} that is a local model for singularities at p of some Shimura variety with level structure of Iwahori type at p. Locally for the étale topology, M(G, μ) is isomorphic to a natural ℤ_{p}-model M(G, μ) of the Shimura variety.

The semi-simple trace of the Frobenius endomorphism on the nearby cycles of M(G, μ) plays an important role in the computation of the local factor at p of the semi-simple Hasse-Weil zeta function of the Shimura variety, see [17]. We can recover the semi-simple trace of Frobenius on the nearby cycles of M(G, μ) from that of the local model M(G, μ), see loc.cit. Thus the problem to calculate the function

x ∈ M(G, μ)(𝔽_{q}) ↦ Tr^{ss}(Fr_{q}, RΨ(ℚ̄_{ℓ})_{x})

comes naturally. R. Kottwitz has conjectured an explicit formula for this function.

To state this conjecture, we note that the set of 𝔽_{q}-points of M(G, μ) can be naturally embedded as a finite set of Iwahori-orbits in the affine flag variety of G(𝔽_{q}((t)))

M(G, μ)(𝔽_{q}) ⊂ G(𝔽_{q}((t)))/I

where I is the standard Iwahori subgroup of G(𝔽_{q}((t))).

Conjecture (Kottwitz) For all x ∈ M(G, μ)(𝔽_{q}),

Tr^{ss}(Fr_{q}, RΨ(ℚ̄_{ℓ})_{x}) = q^{⟨ρ,μ⟩}z_{μ}(x).

Here q^{⟨ρ,μ⟩}z_{μ}(x) is the unique function in the center of the Iwahori-Hecke algebra of I-bi-invariant functions with compact support in G(𝔽_{p}((t))), characterized by

q^{⟨ρ,μ⟩}z_{μ}(x) ∗ 𝕀_{K} = 𝕀_{KμK}.

Here K denotes the maximal compact subgroup G(𝔽_{q}[[t]]) and 𝕀_{KμK} denotes the characteristic function of the double-coset corresponding to a coweight μ.

Kottwitz’ conjecture was first proved for the local model of a special type of Shimura variety with Iwahori type reduction at p attached to the group GL(d) and minuscule coweight (1, 0^{d−1}) (the “Drinfeld case”) in [9]. The method of that paper was one of direct computation: Rapoport had computed the function Tr^{ss}(Fr_{q}, RΨ(ℚ̄_{ℓ})_{x}) for the Drinfeld case (see [17]), and so the result followed from a comparison with an explicit formula for the Bernstein function z_{(1,0^{d−1})}. More generally, the explicit formula for z_{μ} in [9] is valid for any minuscule coweight μ of any quasisplit p-adic group. Making use of this formula, U. Görtz verified Kottwitz’ conjecture for a similar Iwahori-type Shimura variety attached to G = GL(4) and μ = (1, 1, 0, 0), by computing the function Tr^{ss}(Fr_{q}, RΨ(ℚ̄_{ℓ})_{x}) for x ranging over all 33 strata of the corresponding local model M(G, μ).

Shortly thereafter, A. Beilinson and D. Gaitsgory were motivated by Kottwitz’ conjecture to attempt to produce all elements in the center of the Iwahori-Hecke algebra geometrically, via a nearby cycle construction. For this they used Beilinson’s deformation of the affine Grassmannian: a space over a curve X whose fiber over a fixed point x ∈ X is the affine flag variety of the group G, and whose fiber over every other point of X is the affine Grassmannian of G. In [5] Gaitsgory proved a key commutativity result (similar to our Proposition 21) which is valid for any split group G and any dominant coweight, in the function field setting. His result also implies that the semi-simple trace of Frobenius on nearby cycles (of a K-equivariant perverse sheaf on the affine Grassmannian) corresponds to a function in the center of the Iwahori-Hecke algebra of G.

The purpose of this article is to give a proof of Kottwitz’ conjecture for the cases G = GL(d) and G = GSp(2d). In fact we prove a stronger result (Theorem 11) which applies to arbitrary coweights, and which was also conjectured by Kottwitz (although only the case of minuscule coweights seems to be directly related to Shimura varieties).

Main Theorem Let G be either GL(d) or GSp(2d). Then for any dominant coweight μ of G, we have

Tr^{ss}(Fr_{q}, RΨ^{M}(A_{μ,η})) = (−1)^{2⟨ρ,μ⟩} ∑_{λ≤μ} m_{μ}(λ)z_{λ}.

Here M is a member of an increasing family of schemes M_{n_{±}} which contains the local models of Rapoport-Zink; the generic fiber of M can be embedded in the affine Grassmannian of G, and A_{μ,η} denotes the K-equivariant intersection complex corresponding to μ. The special fiber of M embeds in the affine flag variety of G(𝔽̄_{q}((t))) so we can think of the semi-simple trace of Frobenius on nearby cycles as a function in the Iwahori-Hecke algebra of G.

The crucial step in the proof of the theorem is to show that the function Tr^{ss}(Fr_{q}, RΨ^{M}(A_{μ,η})) is in the center of the Iwahori-Hecke algebra. The basic strategy to prove this is:

1. give a geometric construction of convolution of sheaves which corresponds to the usual product in the Hecke algebra,

2. show that convolution commutes with the nearby cycle functor,

3. show that on the generic fiber, convolution of appropriate sheaves is commutative.

While the strategy of proof is similar to that of Beilinson and Gaitsgory, in order to get a statement which is valid over all local non-Archimedean fields we use a somewhat different model, based on spaces of lattices, in the construction of the schemes M_{n_{±}} (we have not determined the precise relation between our model and that of Beilinson-Gaitsgory). This is necessary to compensate for the lack of an adequate notion of affine Grassmannian over p-adic fields. The union of the schemes M_{n_{±}} can be thought of as a p-adic analogue of Beilinson’s deformation of the affine Grassmannian.

We would like to thank G. Laumon and M. Rapoport for generous advice and encouragement. We would like to thank A. Genestier who has pointed out to us a mistake occurring in the first version of this paper. We thank R. Kottwitz for explaining the argument of Beilinson and Gaitsgory to us and for helpful conversations about this material. The first author thanks U. Görtz for helpful conversations. Thanks are also due to the referee for his very careful reading and for his numerous suggestions to correct and improve the manuscript.

T. Haines acknowledges the hospitality and support of the Institut des Hautes Études Scientifiques in Bures-sur-Yvette in the spring of 1999, when this work was begun. He is partially supported by an NSF-NATO Postdoctoral fellowship, and an NSERC Research grant.

B.C. Ngô was visiting the Max Planck Institut fuer Mathematik during the preparation of this article.

2 Rapoport-Zink local models

2.1 Some definitions in the linear case

Let F be a local non-Archimedean field. Let O denote the ring of integers of F and let k = 𝔽_{q} denote the residue field of O. We choose a uniformizer ϖ of O. We denote by η the generic point of S = Spec (O) and by s its closed point.

For G = GL(d) and for μ the minuscule coweight

(\underbrace{1, …, 1}_{r}, \underbrace{0, …, 0}_{d−r})

with 1 ≤ r ≤ d−1, the local model M_{μ} represents the functor which associates to each O-algebra R the set of L_{•} = (L_{0}, …, L_{d−1}) where L_{0}, …, L_{d−1} are R-submodules of R^{d} satisfying the following properties • L_{0}, …, L_{d−1} are locally direct factors of corank r in R^{d},

• α′(L_{0}) ⊂ L_{1}, α′(L_{1}) ⊂ L_{2}, …, α′(L_{d−1}) ⊂ L_{0} where α is the matrix

α′ = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ ϖ & & & 0 \end{matrix} )

The projective S-scheme M_{μ} is a local model for singularities at p of some Shimura variety for unitary group with level structure of Iwahori type at p (see [17],[18]).

Following a suggestion of G. Laumon, we introduce a new variable t and rewrite the moduli problem of M_{μ} as follows. Let M_{μ}(R) be the set of L_{•} = (L_{0}, …, L_{d−1}) where L_{0}, …, L_{d−1} are R[t]-submodules of R[t]^{d}/tR[t]^{d} satisfying the following properties

• as R-modules, L_{0}, …, L_{d−1} are locally direct factors of corank r in R[t]^{d}/tR[t]^{d},

• α(L_{0}) ⊂ L_{1}, α(L_{1}) ⊂ L_{2}, …, α(L_{d−1}) ⊂ L_{0} where α is the matrix

α = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ t + ϖ & & & 0 \end{matrix} )

Obviously, these two descriptions are equivalent because t acts as 0 on the quotient R[t]^{d}/tR[t]^{d}. Nonetheless, the latter description indicates how to construct larger S-schemes M_{μ}, where μ runs over a certain cofinal family of dominant (nonminuscule) coweights.

Let n_{−} ≤ 0 < n_{+} be two integers.

Definition 1 Let M_{r,n_{±}} be the functor which associates each O-algebra R the set of L_{•} = (L_{0}, …, L_{d−1}) where L_{0}, …, L_{d−1} are R[t]-submodules of

t^{n_{−}}R[t]^{d}/t^{n_{+}}R[t]^{d}

satisfying the following properties • as R-modules, L_{0}, …, L_{d−1} are locally direct factors with rank n_{+}d − r

in t^{n_{−}}R[t]^{d}/t^{n_{+}}R[t]^{d},

• α(L_{0}) ⊂ L_{1}, α(L_{1}) ⊂ L_{2}, …, α(L_{d−1}) ⊂ L_{0}.

This functor is obviously represented by a closed sub-scheme in a product of Grassmannians. In particular, M_{r,n_{±}} is projective over S.

In some cases, it is more convenient to adopt the following equivalent description of the functor M_{r,n_{±}}. Let us consider α as an element of the group

α ∈ GL(d, O[t, t^{−1}, (t + ϖ)^{−1}]).

Let V_{0}, V_{1}, …, V_{d} be the fixed O[t]-submodules of O[t, t^{−1}, (t+ϖ)^{−1}]^{d} defined by

V_{i} = α^{−i}O[t]^{d}.

In particular, we have V_{d} = (t + ϖ)^{−1}V_{0}. Denote by V_{i,R} the tensor V_{i} ⊗_{O} R for any O-algebra R.

Definition 2 Let M_{r,n_{±}} be the functor which associates to each O-algebra R the set of

L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d} = (t + ϖ)^{−1}L_{0})

where L_{0}, L_{1}, … are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{d} satisfying the following conditions

• for all i = 0, …, d − 1, we have t^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ t^{n_{−}}V_{i,R},

• as R-modules, L_{i}/t^{n_{+}}V_{i,R} is locally a direct factor of t^{n_{−}}V_{i,R}/t^{n_{+}}V_{i,R} with rank n_{+}d − r.

By using the isomorphism

α^{i} : t^{n_{−}}V_{i,R}/t^{n_{+}}V_{i,R} ⟶^{∼} t^{n_{−}}R[t]^{d}/t^{n_{+}}R[t]^{d}

we can associate to each sequence L_{•} = (L_{i}) as in Definition 1 of M_{r,n_{±}}, the sequence L_{•} = (L_{i}) as in Definition 2, in such a way that

α^{i}(L_{i}/t^{n_{+}}V_{i,R}) = L_{i}.

This correspondence is clearly bijective. Therefore, the two definitions of the functor M_{r,n_{±}} are equivalent.

It will be more convenient to consider the disjoint union M_{n_{±}} of projective schemes M_{r,n_{±}} for all r for which M_{r,n_{±}} makes sense, namely

M_{n_{±}} = ∐_{dn_{−}≤r≤dn_{+}} M_{r,n_{±}},

instead of each connected component M_{r,n_{±}} individually.

2.2 Group action

Definition 2 permits us to define a natural group action on M_{n_{±}}. Every R[t]-module L_{i} as above is included in

t^{n_{+}}R[t]^{d} ⊂ L_{i} ⊂ t^{n_{−}}(t + ϖ)^{−1}R[t]^{d}.

Let L̄_{i} denote its image in the quotient

V̄_{n_{±},R} = t^{n_{−}}(t + ϖ)^{−1}R[t]^{d}/t^{n_{+}}R[t]^{d}.

Obviously, L_{i} is completely determined by L̄_{i}.

Let V̄_{i} denote the image of V_{i} in V̄_{n_{±}}. We can view V̄_{n_{±}} as the free R-module R^{(n_{+}−n_{−}+1)d} equipped with the endomorphism t and with the filtration

V̄_{•} = (V̄_{0} ⊂ V̄_{1} ⋯ ⊂ V̄_{d} = (t + ϖ)^{−1}V̄_{0})

which is stabilized by t.

We now consider the functor J_{n_{±}} which associates to each O-algebra R the group J_{n_{±}}(R) of all R[t]-automorphisms of V̄_{n_{±}} fixing the filtration V̄_{•}. This functor is represented by a closed subgroup of GL((n_{+} − n_{−} + 1)d) over S that acts in the obvious way on M_{n_{±}}.

Lemma 3 The group scheme J_{n_{±}} is smooth over S.

Proof. Consider the functor J_{n_{±}} which associates to each O-algebra R the ring J_{n_{±}}(R) of all R[t]-endomorphisms of V̄_{n_{±}} stabilizing the filtration V̄_{•}. This functor is obviously represented by a closed sub-scheme of the S-scheme 𝔤𝔩((n_{+} − n_{−} + 1)d) of square matrices with rank (n_{+} − n_{−} + 1)d.

The natural morphism of functors J_{n_{±}} → J_{n_{±}} is an open immersion. Thus it suffices to prove that J_{n_{±}} is smooth over S.

Giving an element of J_{n_{±}} is equivalent to giving d vectors v_{1}, …, v_{d} such that v_{i} ∈ t^{n_{−}} V̄_{i}. This implies that J_{n_{±}} is isomorphic to a trivial vector bundle over S of rank

∑_{i=1}^{d} rk_{O}(t^{n_{−}}V_{i}/t^{n_{+}}O[t]^{d}) = d^{2}(n_{+} − n_{−} + 1) − (d − 1)d/2.

This finishes the proof of the lemma. □

2.3 Description of the generic fiber

For this purpose, we use Definition 1 of M_{n_{±}}. Let R be an F -algebra. The matrix α then is invertible as an element

α ∈ GL(d, R[t]/t^{n_{+}−n_{−}}R[t]),

the group of automorphisms of t^{n_{−}}R[t]^{d}/t^{n_{+}}R[t]^{d}.

Let (L_{0}, …, L_{d−1}) be an element of M_{n_{±}}(R). As R-modules, the L_{i} are locally direct factors of the same rank. For i = 1, …, d − 1, the inclusion α(L_{i−1}) ⊂ L_{i} implies the equality α(L_{i−1}) = L_{i}. In this case, the last inclusion α(L_{d−1}) ⊂ L_{0} is automatically an equality, because the matrix

α^{d} = diag (t + ϖ, …, t + ϖ)

satisfies the property: α^{d}(L_{0}) = L_{0}. In others words, the whole sequence (L_{0}, …, L_{d−1}) is completely determined by L_{0}.

Let us reformulate the above statement in a more precise way. Let Grass_{n_{±}} be the functor which associates to each O-algebra R the set of R[t]-submodules L of t^{n_{−}}R[t]^{d}/t^{n_{+}}R[t]^{d} which, as R-modules, are locally direct factors of t^{n_{−}}R[t]^{d}/t^{n_{+}}R[t]^{d}. Obviously, this functor is represented by a closed subscheme of a disjoint union of Grassmannians. In particular, it is proper over S.

Let π : M_{n_{±}} → Grass_{n_{±}} be the morphism defined by

π(L_{0}, …, L_{d−1}) = L_{0}.

The above discussion can be reformulated as follows.

Lemma 4 The morphism π : M_{n_{±}} → Grass_{n_{±}} is an isomorphism over the generic point η of S. □

Let K_{n_{±}} the functor which associates to each O-algebra R the group

K_{n_{±}} = GL(d, R[t]/t^{n_{+}−n_{−}}R[t]).

Obviously, it is represented by a smooth group scheme over S and acts naturally on Grass_{n_{±}}. This action yields a decomposition into orbits that are smooth over S

Grass_{n_{±}} = ∐_{λ∈Λ(n_{±})} O_{λ}

where Λ(n_{±}) is the finite set of sequences of integers λ = (λ_{1}, …, λ_{d}) satisfying the following condition

n_{+} ≥ λ_{1} ≥ ⋯ ≥ λ_{d} ≥ n_{−}.

This set Λ(r, n_{±}) can be viewed as a finite subset of the cone of dominant coweights of G = GL(d) and conversely, every dominant coweight of G occurs in some Λ(n_{±}). For all λ ∈ Λ(n_{±}), we have

O_{λ}(F) = K_{F} t^{λ}K_{F} /K_{F} .

Here K_{F} = GL(d, F [[t]]) is the standard maximal “compact” subgroup of G_{F} = GL(d, F ((t))) and acts on Grass_{n_{±}}(F) through the quotient K_{n_{±}}(F). The above equality holds if one replaces F by any field which is also an O-algebra, since K_{n_{±}} is smooth; in particular it holds for the residue field k.

We derive from the above lemma the description

M_{n_{±}}(F) = ∐_{λ∈Λ(n_{±})} K_{F} t^{λ}K_{F} /K_{F} .

We will need to compare the action of J_{n_{±}} on M_{n_{±}} and the action of K_{n_{±}} on Grass_{n_{±}}. By definition, J_{n_{±}}(R) is a subgroup of

J_{n_{±}}(R) ⊂ GL(d, R[t]/t^{n_{+}−n_{−}}(t + ϖ)R[t])

for any O-algebra R. By using the natural homomorphism

GL(d, R[t]/t^{n_{+}−n_{−}}(t + ϖ)R[t]) → GL(d, R[t]/t^{n_{+}−n_{−}}R[t])

we get a homomorphism J_{n_{±}}(R) → K_{n_{±}}(R). This gives rises to a homomorphism of group schemes ρ : J_{n_{±}} → K_{n_{±}}, which is surjective over the generic point η of S.

The proof of the following lemma is straightforward.

Lemma 5 With respect to the homomorphism ρ : J_{n_{±}} → K_{n_{±}}, and to the morphism π : M_{n_{±}} → Grass_{n_{±}}, the action of J_{n_{±}} on M_{n_{±}} and the action of K_{n_{±}} on Grass_{n_{±}} are compatible. □

2.4 Description of the special fiber

For this purpose, we will use Definition 2 of M_{n_{±}}. The functor M_{r,n_{±}} associates to each k-algebra R the set of

L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d} = t^{−1}L_{0})

where L_{0}, L_{1}, … are R[t]-submodules of R[t, t^{−1}]^{d} satisfying the following conditions

• for all i = 0, …, d − 1, we have t^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ t^{n_{−}}V_{i,R},

• as an R-module, each L_{i}/t^{n_{+}}V_{i,R} is locally a direct factor of t^{n_{−}}V_{i,R}/t^{n_{+}}V_{i,R} with rank n_{+}d − r.

Let I_{k} denote the standard Iwahori subgroup of G_{k} = GL(d, k((t))), that is, the subgroup of integer matrices GL(d, k[[t]]) whose reduction mod t lies in the subgroup of upper triangular matrices in GL(d, k). The set of k-points of M_{n_{±}} can be realized as a finite subset in the set of affine flags of GL(d)

M_{n_{±}}(k) ⊂ G_{k}/I_{k}.

By definition, the k-points of J_{n_{±}} are the matrices in GL(d, k[t]/t^{n_{+}−n_{−}+1}k[t]) whose reduction mod t is upper triangular. Thus, J_{n_{±}}(k) is a quotient of I_{k}. Obviously, the action of J_{n_{±}}(k) on M_{n_{±}}(k) and the action of I_{k} on G_{k}/I_{k} are compatible. Therefore, for each r such that dn_{−} ≤ r ≤ dn_{+} there exists a finite subset W̃ (r, n_{±}) ⊂ W̃ of the affine Weyl group W̃ such that

M_{n_{±}}(k) = ∐_{w∈W̃ (n_{±})} I_{k}wI_{k}/I_{k},

where W̃ (n_{±}) = ∐_{r} W̃ (r, n_{±}). One can see easily that any element w ∈ W̃ occurs in the finite subset W̃ (n_{±}) for some n_{±}. But the exact determination of the finite sets W̃ (r, n_{±}) is a difficult combinatorial problem; for the case of minuscule coweights of GL(d) (i.e., n_{+} = 1 and n_{−} = 0) these sets have been described by Kottwitz and Rapoport [13]. ^{1}

Let us recall that

Grass_{n_{±}}(k) = ∐_{λ∈Λ(n_{±})} K_{k} t^{λ}K_{k}/K_{k}.

The proof of the next lemma is straightforward.

Lemma 6 The map π(k) : M_{n_{±}}(k) → Grass_{n_{±}}(k) is the restriction of the natural map G_{k}/I_{k} → G_{k}/K_{k}.

2.5 Symplectic case

For the symplectic case, we will give only the definitions of the symplectic analogies of the objects which were considered in the linear case. The statements of Lemmas 3,4,5 and 6 remain unchanged.

In this section, the group G stands for GSp(2d) associated to the symplectic form ⟨ , ⟩ represented by the matrix

( \begin{matrix}0 & J \\ −J & 0 \end{matrix} )

where J is the anti-diagonal matrix with entries equal to 1. Let μ denote the minuscule coweight

μ = (\underbrace{1, …, 1}_{d}, \underbrace{0, …, 0}_{d}).

Following Rapoport and Zink ([18]) the local model M_{μ} represents the functor which associates to each O-algebra R the set of sequences L_{•} = (L_{0}, …, L_{d}) where L_{0}, …, L_{d} are R-submodules of R^{2d} satisfying the following properties

• L_{0}, …, L_{d} are locally direct factors of R^{2d} of rank d, • α′(L_{0}) ⊂ L_{1}, …, α′(L_{d−1}) ⊂ L_{d} where α′ is the matrix of size 2d × 2d

1We refer to our subsequent work [10] for further progress in the description of the sets W̃ (r, n_{±}). In the terminology of Kottwitz-Rapoport [13], the set W̃ (r, n_{±}) is precisely the set of μ-permissible elements, for μ = (n_{+}^{q}, R + n_{−}, n_{−}^{d−q−1}), where q and R are defined by r − dn_{−} = q(n_{+} − n_{−}) + R, with 0 ≤ R < n_{+} − n_{−}. By the main result of [10], it is also the set of μ-admissible elements. Similar remarks apply to the sets W̃ (n_{±}) occurring in the symplectic case, cf. end of section 2.5.

α′ = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ ϖ & & & 0 \end{matrix} )

• L_{0} and L_{d} are isotropic with respect to ⟨ , ⟩.

Just as in the linear case, let us introduce a new variable t and give the symplectic analogue of Definition 2. We consider the matrix of size 2d × 2d

α′ = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ t + ϖ & & & 0 \end{matrix} )

viewed as an element of

α ∈ GL(2d, O[t, t^{−1}, (t + ϖ)^{−1}]).

Denote by V_{0}, …, V_{2d−1} the fixed O[t]-submodules of O[t, t^{−1}, (t + ϖ)^{−1}]^{2d} defined by V_{i} = α^{−i}O[t]^{2d}. For an O-algebra R, let V_{i,R} denote V_{i} ⊗_{O} R.

For any R[t]-submodule L of R[t, t^{−1}, (t + ϖ)^{−1}]^{2d}, the R[t]-module

L^{⊥′} = {x ∈ R[t, t^{−1}, (t + ϖ)^{−1}]^{2d} | ∀y ∈ L, t^{n}(t + ϖ)^{n′}⟨x, y⟩ ∈ R[t]}

is called the dual of L with respect to the form ⟨ , ⟩′ = t^{n}(t + ϖ)^{n′}⟨ , ⟩. Thus V_{0} is autodual with respect to the form ⟨ , ⟩ and V_{d} is autodual with respect to the form (t + ϖ)⟨ , ⟩.

Here is the symplectic analogue of Definition 2 of the model M_{n_{±}}. For n_{−} = 0 and n_{+} = 1, M_{n_{±}} will coincide with M_{μ}, for μ = (1^{d}, 0^{d}):

Definition 7 For any n_{−} ≤ 0 < n_{+}, let M_{n_{±}} be the functor which associates to each O-algebra R the set of sequences

L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d})

where L_{0}, …, L_{d} are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{2d} satisfying the following properties • for all i = 0, …, d, we have t^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ t^{n_{−}}V_{i,R},

• as R-modules, L_{i}/t^{n_{+}}V_{i,R} is locally a direct factor of t^{n_{−}}V_{i,R}/t^{n_{+}}V_{i,R} of rank (n_{+} − n_{−})d,

• L_{0} is autodual with respect to the form t^{−n_{−}−n_{+}}⟨ , ⟩, and L_{d} is autodual with respect to the form t^{−n_{−}−n_{+}}(t + ϖ)⟨ , ⟩.

Let us now define the natural group action on M_{n_{±}}. The functor J_{n_{±}} associates to each O-algebra R the group J_{n_{±}}(R) of R[t]-linear automorphisms of

V̄_{n_{±},R} = t^{n_{−}}(t + ϖ)^{−1}R[t]^{2d}/t^{n_{+}}R[t]^{2d}

which fix the filtration

V̄_{•,R} = (V̄_{0,R} ⊂ ⋯ ⊂ V̄_{d,R})

(the image of V_{•,R} in V̄_{n_{±},R}) and which fix, up to a unit in R, the symplectic form t^{−n_{−}−n_{+}}(t + ϖ)⟨ , ⟩. This functor is represented by an S-group scheme J_{n_{±}} which acts on M_{n_{±}}. Lemma 3 remains true in the symplectic case : J_{n_{±}} is a smooth group scheme over S. The proof is completely similar to the linear case.

Let us now describe the generic fiber of M_{n_{±}}. Let Grass_{n_{±}} be the functor which associates to each O-algebra R the set of R[t]-submodules L of t^{n_{−}}R[t]^{2d}/t^{n_{+}}R[t]^{2d} which, as R-modules, are locally direct factors of rank (n_{+} − n_{−})d and which are isotropic with respect to t^{−n_{−}−n_{+}}⟨ , ⟩. Then the morphism π : M_{n_{±}} → Grass_{n_{±}} defined by π(L_{•}) = L_{0} is an isomorphism over the generic point η of S. Let K_{n_{±}} denote the functor which associates to each O-algebra R the group of R[t]-automorphisms of t^{n_{−}}R[t]^{2d}/t^{n_{+}}R[t]^{2d} which fix the symplectic form t^{−n_{−}−n_{+}}⟨ , ⟩ up to a unit in R. Then K_{n_{±}} is represented by a smooth group scheme over S, and it acts in the obvious way on Grass_{n_{±}}. Consequently, we have a stratification in orbits of the generic fiber M_{n_{±},η}

M_{n_{±},η} = ∐_{λ∈Λ(n_{±})} O_{λ,η}.

Here Λ(n_{±}) is the set of sequences λ = (λ_{1}, …, λ_{d}) satisfying

n_{+} ≥ λ_{1} ≥ ⋯ ≥ λ_{d} ≥ \frac{n_{+} + n_{−}}{2} ,

and can be viewed as finite subset of the cone of dominant coweights of G = GSp(2d). One can easily check that each dominant coweight of GSp(2d) occurs in some Λ(n_{±}). For any λ ∈ Λ(n_{±}), we have also

O_{λ,η}(F) = K_{F} t^{λ}K_{F} /K_{F}

where K_{F} = G(F [[t]]) is the ”maximal compact” subgroup of G_{F} = G(F ((t))).

Next we turn to the special fiber of M_{n_{±}}. For this it is most convenient to give a slight reformulation of Definition 7 above. Let R be any O-algebra. It is easy to see that specifying a sequence L_{•} = (L_{0} ⊂ … L_{d}) as in Definition 7 is the same as specifying a periodic “lattice chain”

… ⊂ L_{−1} ⊂ L_{0} ⊂ … ⊂ L_{2d} = (t + ϖ)^{−1}L_{0} ⊂ …

consisting of R[t]-submodules of R[t, t^{−1}, (t+ϖ)^{−1}]^{2d} with the following properties:

• t^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ t^{n_{−}}V_{i,R}, where V_{i,R} = α^{−i}V_{0,R}, for every i ∈ ℤ,

• L_{i}/t^{n_{+}}V_{i,R} is locally a direct factor of rank (n_{+} − n_{−})d, for every i ∈ ℤ,

• L_{i}^{⊥} = t^{−n_{−}−n_{+}}L_{−i}, for every i ∈ ℤ,

where ⊥ is defined using the original symplectic form ⟨ , ⟩ on R[t, t^{−1}, (t + ϖ)^{−1}]^{2d}. We denote by I_{k} the standard Iwahori subgroup of GSp(2d, k[[t]]), namely, the stabilizer in this group of the periodic lattice chain V_{•,k[[t]]}. There is a canonical surjection I_{k} → J_{n_{±}}(k) and so the Iwahori subgroup I_{k} acts via its quotient J_{n_{±}}(k) on the set M_{n_{±}}(k). Moreover, the I_{k}-orbits in M_{n_{±}}(k) are parametrized by a certain finite set W̃ (n_{±}) of the affine Weyl group W̃ (GSp(2d))

M_{n_{±}}(k) = ∐_{w∈W̃ (n_{±})} I_{k} w I_{k}/I_{k}.

The precise description of the sets W̃ (n_{±}) is a difficult combinatorial problem (see [13] for the case n_{+} = 1, n_{−} = 0), but one can easily see that any w ∈ W̃ (GSp(2d)) is contained in some W̃ (n_{±}).

The definitions of the group scheme action of K_{n_{±}} on Grass_{n_{±}}, of the homomorphism ρ : J_{n_{±}} → K_{n_{±}} and the compatibility properties (Lemmas 5,6) are obvious and will be left to the reader.

3 Semi-simple trace on nearby cycles

3.1 Semi-simple trace

The notion of semi-simple trace was introduced by Rapoport in [17] and its good properties were mentioned there. The purpose of this section is only to give a more systematic presentation in insisting on the important fact that the semi-simple trace furnish a kind of sheaf-function dictionary à la Grothendieck. In writing this section, we have benefited from very helpful explanations of Laumon.

Let F̄ be a separable closure of the local field F. Let Γ be the Galois group Gal(F̄/F) of F and let Γ_{0} be the inertia subgroup of Γ defined by the exact sequence

1 → Γ_{0} → Γ → Gal(k̄/k) → 1.

For any prime ℓ ≠ p, there exists a canonical surjective homomorphism

t_{ℓ} : Γ_{0} → ℤ_{ℓ}(1).

Let R denote the abelian category of continuous, finite dimensional ℓ-adic representations of Γ. Let (ρ, V) be an object of R

ρ : Γ → GL(V).

According to a theorem of Grothendieck, 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 (the residue field k is supposed finite). There exists then an unique nilpotent morphism, the logarithm of ρ

N : V (1) → V

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

ρ(g) = exp(Nt_{ℓ}(g)).

Following Rapoport, an increasing filtration F of V will be called admissible if it is stable under the action of Γ and such that Γ_{0} operates on the associated graded gr_{•}^{F}(V) through a finite quotient. Admissible filtrations always exist: we can take for instance the filtration defined by the kernels of the powers of N.

We define the semi-simple trace of Frobenius on V as

Tr^{ss}(Fr_{q}, V) = ∑_{k} Tr(Fr_{q}, gr_{k}^{F}(V)^{Γ_{0}}).

Lemma 8 The semi-simple trace Tr^{ss}(Fr_{q}, V) does not depend on the choice of the admissible filtration F.

Proof. Let us first consider the case where Γ_{0} acts on V through a finite quotient. Since taking invariants under a finite group acting on a ℚ̄_{ℓ}-vector space is an exact functor, the graded associated to the filtration F′ of V^{Γ_{0}} induced by F is equal to gr_{•}^{F}(V)^{Γ_{0}}

gr_{k}^{F′}(V^{Γ_{0}}) = gr_{k}^{F}(V)^{Γ_{0}}.

Consequently

Tr(Fr_{q}, V^{Γ_{0}}) = ∑_{k} Tr(Fr_{q}, gr_{k}^{F}(V)^{Γ_{0}}).

In the general case, any two admissible filtrations admit a third finer admissible filtration. By using the above case, one sees the semi-simple trace associated to each of the two first admissible filtrations is equal to the semisimple trace associated to the third one and the lemma follows. □

Corollary 9 The function defined by

V ↦ Tr^{ss}(Fr_{q}, V)

on the set of isomorphism classes V of R, factors through the Grothendieck group of R.

For any object C of the derived category associated to R, we put

Tr^{ss}(Fr_{q}, C) = ∑_{i}(−1)^{i}Tr^{ss}(Fr_{q}, H^{i}(C)).

By the above corollary, for any distinguished triangle

C → C′ → C′′ → C[1]

the equality

Tr^{ss}(Fr_{q}, C) + Tr^{ss}(Fr_{q}, C′′) = Tr^{ss}(Fr_{q}, C′)

holds.

Let X be a k-scheme of finite type, Xs̄ = X ⊗_{k} k̄. Let D_{c}^{b}(X ×_{k} η) denote the derived category associated to the abelian category of constructible ℓadic sheaves on Xs̄ equipped with an action of Γ compatible with the action of Γ on Xs̄ through Gal(k̄/k), see [3]. ^{2} Let C be an object of D_{c}^{b}(X ×_{k} η). For any x ∈ X(k), the fiber C_{x} is an object of the derived category of R. Thus we can define the function semi-simple trace

τ_{C}^{ss} : X(k) → ℚ̄_{ℓ}

by

τ_{C}^{ss}(x) = Tr^{ss}(Fr_{q}, C_{x}).

This association C ↦ τ_{C}^{ss} furnishes an analogue of the usual sheaf-function dictionary of Grothendieck (see [7]):

Proposition 10 Let f : X → Y be a morphism between k-schemes of finite type.

1. Let C be an object of D_{c}^{b}(Y ×_{k} η). For all x ∈ X(k), we have

τ_{f^{∗}C}^{ss}(x) = τ_{C}^{ss}(f(x))

2. Let C be an object of D_{c}^{b}(X ×_{k} η). For all y ∈ Y (k), we have

τ_{Rf_{!}C}^{ss}(y) = ∑_{{x∈X(k) \atop f(x)=y}} τ_{C}^{ss}(x).

Proof. The first statement is obvious because f^{∗}C_{x} and C_{f(x)} are canonically isomorphic as objects of the derived category of R.

It suffices to prove the second statement in the case Y = s. By Corollary 9 and “shifting”, it suffices to consider the case where C is concentrated in only one degree, say in the degree zero. Denote C = H^{0}(C) and choose an admissible filtration of C

0 = C_{0} ⊂ C_{1} ⊂ C_{2} ⊂ ⋯ ⊂ C_{n} = C.

2The category D_{c}^{b}(X ×_{k} η) is defined, following [4], to be ℚ_{ℓ}⊗ the projective 2-limit of the categories D_{ctf}^{b}(X ×_{k} η, ℤ/ℓ^{n}ℤ), so it is not strictly speaking the derived category of the abelian category of constructible ℓ-adic sheaves.

The associated spectral sequence

E_{1}^{i,j−i} = H_{c}^{j}(Xs̄, C_{i}/C_{i−1}) ⟹ H_{c}^{j}(Xs̄, C)

yields an abutment filtration on H_{c}^{j}(Xs̄, C) with associated graded E_{∞}^{i,j−i}. Since the inertia group acts on E_{1}^{i,j−i} through a finite quotient, the same property holds for E_{∞}^{i,j−i} because E_{∞}^{i,j−i} is a subquotient of E_{1}^{i,j−i}. Consequently, the abutment filtration on H_{c}^{j}(Xs̄, C) is an admissible filtration and by definition, we have

Tr^{ss}(Fr_{q}, Rf_{!}C) = ∑_{i,j}(−1)^{j}Tr(Fr_{q}, (E_{∞}^{i,j−i})^{Γ_{0}}).

Now, the identity in the Grothendieck group

∑_{i,j}(−1)^{j}E_{1}^{i,j−i} = ∑_{i,j}(−1)^{j}E_{∞}^{i,j−i}

implies

∑_{i,j}(−1)^{j}(E_{1}^{i,j−i})^{Γ_{0}} = ∑_{i,j}(−1)^{j}(E_{∞}^{i,j−i})^{Γ_{0}}

because taking the invariants under a finite group is an exact functor. The same exactness implies

(E_{1}^{i,j−i})^{Γ_{0}} = H_{c}^{j}(Xs̄, C_{i}/C_{i−1})^{Γ_{0}} = H_{c}^{j}(Xs̄, (C_{i}/C_{i−1})^{Γ_{0}}).

By putting the above equalities together, we obtain

Tr^{ss}(Fr_{q}, Rf_{!}C) = ∑_{i,j}(−1)^{j}Tr(Fr_{q}, H_{c}^{j}(Xs̄, (C_{i}/C_{i−1})^{Γ_{0}})).

By using now the Grothendieck-Lefschetz formula, we have

∑_{x∈X(k)} Tr(Fr_{q}, (C_{i}/C_{i−1})_{x}^{Γ_{0}}) = ∑_{j}(−1)^{j}Tr(Fr_{q}, H_{c}^{j}(Xs̄, (C_{i}/C_{i−1})^{Γ_{0}})).

Consequently,

Tr^{ss}(Fr_{q}, Rf_{!}C) = ∑_{x∈X(k)} Tr^{ss}(Fr_{q}, C_{x}). □

3.2 Nearby cycles

Let η̄ = Spec (F̄) denote the geometric generic point of S, S̄ be the normalization of S in η̄ and s̄ be the closed point of S̄. For an S-scheme X of finite type, let us denote by j̄^{X} : Xη̄ → XS̄ the morphism deduced by base change from j̄: η̄ → S̄ and denote by ī^{X} : Xs̄ → XS̄ the morphism deduced from ī: s̄ → S̄.

The nearby cycles of an ℓ-adic complex C_{η} on X_{η}, is the complex of ℓ-adic sheaves defined by

RΨ^{X}(C_{η}) = i^{X,∗}Rj̄_{∗}^{X}j̄^{X,∗}C_{η}.

The complex RΨ^{X}(C_{η}) is equipped with an action of Γ compatible with the action of Γ on Xs̄ through the quotient Gal(k̄/k).

For X a proper S-scheme, we have a canonical isomorphism

RΓ(Xs̄, RΨ(C_{η})) = RΓ(Xη̄, C_{η})

compatible with the natural actions of Γ on the two sides.

Let us suppose moreover the generic fiber X_{η} is smooth. In order to compute the local factor of the Hasse-Weil zeta function, one should calculate the trace

∑_{j}(−1)^{j}Tr(Fr_{q}, H^{j}(Xη̄, ℚ̄_{ℓ})^{Γ_{0}}).

Assuming that the graded pieces in the monodromy filtration of H^{j}(Xη̄, ℚ̄_{ℓ}) are pure (Deligne’s conjecture), Rapoport proved that the true local factor is completely determined by the semi-simple local factor, see [17]. Now by the above discussion the semi-simple trace can be computed by the formula

∑_{j}(−1)^{j}Tr^{ss}(Fr_{q}, H^{j}(Xη̄, ℚ̄_{ℓ})) = ∑_{x∈X(k)} Tr^{ss}(Fr_{q}, RΨ(ℚ̄_{ℓ})_{x}).

4 Statement of the main result

4.1 Nearby cycles on local models

We have seen in subsection 2.3 (resp. 2.5 for symplectic case) that the generic fiber of M_{n_{±}} admits a stratification with smooth strata

M_{n_{±},η} = ∐_{λ∈Λ(n_{±})} O_{λ,η}.

Denote by Ō_{λ,η} the Zariski closure of O_{λ,η} in M_{n_{±},η}; in general Ō_{λ,η} is no longer smooth . It is natural to consider A_{λ,η} = IC(O_{λ,η}), its ℓ-adic intersection complex.

We want to calculate the function

τ_{RΨ^{M} (A_{λ,η})}^{ss}(x) = Tr^{ss}(Fr_{q}, RΨ^{M}(A_{λ,η})_{x})

of semi-simple trace of the Frobenius endomorphism on the nearby cycle complex RΨ^{M}(A_{λ,η}) defined in the last section. We are denoting the scheme M_{n_{±}} simply by M here.

As O_{λ,η} is an orbit of J_{n_{±},η}, the intersection complex A_{λ,η} is naturally J_{n_{±},η}-equivariant. As we know that J_{n_{±}} is smooth over S by Lemma 3, its nearby cycle complex RΨ^{M}(A_{λ,η}) is J_{n_{±},s̄}-equivariant. In particular, the function

τ_{RΨ^{M} (A_{λ,η})}^{ss} : M_{n_{±}}(k) → ℚ̄_{ℓ}

is J_{n_{±}}(k)-invariant.

Now following the group theoretic description of the action of J_{n_{±}}(k) on M_{n±}(k) in subsection 2.4 (resp. 2.5), we can consider the function τ_{RΨ^{M} (A_{λ,η})}^{ss} as a function on G_{k} with compact support which is invariant on the left and on the right by the Iwahori subgroup I_{k}

τ_{RΨ^{M} (A_{λ,η})}^{ss} ∈ H(G_{k}//I_{k}).

The following statement was conjectured by R. Kottwitz, and is the main result of this paper.

Theorem 11 Let G be either GL(d) or GSp(2d). Let M = M_{n_{±}} be the scheme associated to the group G and the pair of integers n_{±}, as above. Then we have the formula

τ_{RΨ^{M} (A_{λ,η})}^{ss} = (−1)^{2⟨ρ,λ⟩} ∑_{λ′≤λ} m_{λ}(λ′)z_{λ′}

where z_{λ′} is the function of Bernstein associated to the dominant coweight λ′, which lies in the center Z(H(G_{k}//I_{k})) of H(G_{k}//I_{k}).

Here, ρ is half the sum of positive roots for G and thence 2⟨ρ, λ⟩ is the dimension of O_{λ,η}. The integer m_{λ}(λ′) is the multiplicity of weight λ′ occurring in the representation of highest weight λ. The partial ordering λ′ ≤ λ is defined to mean that λ − λ′ is a sum of positive coroots of G.

Comparing with the formula for minuscule μ given in Kottwitz’ conjecture (cf. Introduction), one notices the absence of the factor q^{⟨ρ,μ⟩} and the appearance of the sign (−1)^{2⟨ρ,μ⟩}. This difference is explained by the normalization of the intersection complex A_{μ,η}. For minuscule coweights μ, the orbit O_{μ} is closed. Consequently, the intersection complex A_{μ,η} differs from the constant sheaf only by a normalization factor

A_{μ,η} = ℚ̄_{ℓ}[2⟨ρ, μ⟩](⟨ρ, μ⟩).

We refer to Lusztig’s article [14] for the definition of Bernstein’s functions. In fact, what we need is rather the properties that characterize these functions. We will recall these properties in the next subsection.

4.2 The Satake and Bernstein isomorphisms

Denote by K_{k} the standard maximal compact subgroup G(k[[t]]) of G_{k}, where G is either GL(d) or GSp(2d). The ℚ̄_{ℓ}-valued functions with compact support in G_{k} invariant on the left and on the right by K_{k} form a commutative algebra H(G_{k}//K_{k}) with respect to the convolution product. Here the convolution is defined using the Haar measure on G_{k} which gives K_{k} measure 1. Denote by 𝕀_{K} the characteristic function of K_{k}. This element is the unit of the algebra H(G_{k}//K_{k}). Similarly we define the convolution on H(G_{k}//I_{k}) using the Haar measure on G_{k} which gives I_{k} measure 1.

We consider the following triangle

ℚ̄_{ℓ}[X_{∗}]^{W}
Bern. ↙  ↖^{Sat.}
Z(H(G_{k}//I_{k})) ⟶_{−∗𝕀_{K}} H(G_{k}//K_{k})

Here ℚ̄_{ℓ}[X_{∗}]^{W} is the W -invariant sub-algebra of the ℚ̄_{ℓ}-algebra associated to the group of cocharacters of the standard (diagonal) torus T in G and W is the Weyl group associated to T. For the case G = GL(d), this algebra is isomorphic to the algebra of symmetric polynomials with d variables and their inverses: ℚ̄_{ℓ}[X_{1}^{±}, …, X_{d}^{±}]^{S_{d}}.

The above maps

Sat : H(G_{k}//K_{k}) → ℚ̄_{ℓ}[X_{∗}]^{W}

and

Bern : ℚ̄_{ℓ}[X_{∗}]^{W} → Z(H(G_{k}//I_{k}))

are the isomorphisms of algebras constructed by Satake, see [19] and by Bernstein, see [14]. It follows immediately from its definition that the Bernstein isomorphism sends the irreducible character χ_{λ} of highest weight λ to

Bern(χ_{λ}) = ∑_{λ′≤λ} m_{λ}(λ′)z_{λ′}.

The horizontal map

Z(H(G_{k}//I_{k})) → H(G_{k}//K_{k})

is defined by f ↦ f ∗ 𝕀_{K} where

f ∗ 𝕀_{K}(g) = ∫_{G_{k}} f(gh^{−1})𝕀_{K}(h) dh.

The next statement seems to be known to the experts. It can be deduced easily, see [8], from results of Lusztig [14] and Kato [12]. Another proof can be found in an article of Dat [2].

Lemma 12 The above triangle is commutative.

It follows that the horizontal map is an isomorphism, and that (−1)^{2⟨ρ , λ⟩} ∑_{λ′≤λ} m_{λ}(λ′)z_{λ′} is the unique element in Z(H(G_{k}//I_{k})) whose image in H(G_{k}//K_{k}) has Satake transform (−1)^{2⟨ρ , λ⟩}χ_{λ}.

Thus in order to prove the Theorem 11, it suffices now to prove the two following statements.

Proposition 13 The function τ_{RΨ^{M} (A_{λ,η})}^{ss} lies in the center Z(H(G_{k}//I_{k})) of the algebra H(G_{k}//I_{k}).

Proposition 14 The Satake transform of τ_{RΨ^{M} (A_{λ,η})}^{ss}∗𝕀_{K} is equal to (−1)^{2⟨ρ,λ⟩}χ_{λ}, where χ_{λ} is the irreducible character of highest weight λ.

In fact we can reformulate Proposition 14 in such a way that it becomes independent of Proposition 13. We will prove Proposition 14 in the next section.

In order to prove Proposition 13, we have to adapt Lusztig’s construction of geometric convolution to our context. This will be done in the section 7. The proof of Proposition 13 itself will be given in section 8.

5 Proof of Proposition 14

5.1 Averaging by K

The map

Z(H(G_{k}//I_{k})) → H(G_{k}//K_{k})

defined by f ↦ f ∗ 𝕀_{K} can be obviously extended to a map

C_{c}(G_{k}/I_{k}) → C_{c}(G_{k}/K_{k})

where C_{c}(G_{k}/I_{k}) (resp. C_{c}(G_{k}/K_{k})) is the space of functions with compact support in G_{k} invariant on the right by I_{k} (resp. K_{k}). This map can be rewritten as follows

f ∗ 𝕀_{K}(g) = ∑_{h∈K_{k}/I_{k}} f(gh).

Therefore, this operation corresponds to summing along the fibers of the map G_{k}/I_{k} → G_{k}/K_{k}. For the particular function τ_{RΨ^{M} (A_{λ,η})}^{ss}, it amounts to summing along the fibers of the map

π(k) : M_{n_{±}}(k) → Grass_{n_{±}}(k),

(see Lemma 6).

By using now the sheaf-function dictionary for semi-simple trace, we get

τ_{RΨ^{M} (A_{λ,η})}^{ss} ∗ 𝕀_{K} = τ_{Rπ_{s̄,∗}RΨ^{M} (A_{λ,η})}^{ss}.

The nearby cycle functor commutes with direct image by a proper morphism, so that

Rπ_{s̄,∗}RΨ^{M}(A_{λ,η}) = RΨ^{Grass}Rπ_{η,∗}(A_{λ,η}).

By Lemma 4, π_{η} is an isomorphism. Consequently, Rπ_{η,∗}(A_{λ,η}) = A_{λ,η}.

According to the description of Grass = Grass_{n_{±}} (see subsections 2.3 and 2.5), we can prove that RΨ^{Grass}A_{λ,η} = A_{λ,s̄} (note that the complex A_{λ,η} over Grass_{η} can be extended in a canonical fashion to a complex A_{λ} over the Sscheme Grass, thus A_{λ,s̄} makes sense). In particular, the inertia subgroup Γ_{0} acts trivially on RΨ^{Grass}A_{λ,η} and the semi-simple trace is just the ordinary trace. The proof of a more general statement will be given in the following appendix.

By putting together the above equalities, we obtain

Rπ_{s̄,∗}RΨ^{M}(A_{λ,η}) = A_{λ,s}.

To conclude the proof of Proposition 14, we quote an important theorem of Lusztig and Kato, see [14] and [12]. We remark that Ginzburg and also Mirkovic and Vilonen have put this result in its natural framework : a Tannakian equivalence, see [6],[16].

Theorem 15 (Lusztig, Kato) The Satake transform of the function τ_{A_{λ,s}} is equal to

Sat(τ_{A_{λ,s}}) = (−1)^{2⟨ρ,λ⟩}χ_{λ}

where χ_{λ} is the irreducible character of highest weight λ.

5.2 Appendix

This appendix seems to be well known to the experts. We thank G. Laumon who has kindly explained it to us.

Let us consider the following situation.

Let X be a proper scheme over S equipped with an action of a group scheme J smooth over S. We suppose there is a stratification

X = ∐_{α∈Δ} X_{α}

with each stratum X_{α} smooth over S. We assume that the group scheme J acts transitively on all fibers of X_{α}. Moreover, we suppose there exists, for each α, a J-equivariant resolution of singularities X̃_{α}

π_{α} : X̃_{α} → X̄_{α}

of the closure X̄_{α} of X_{α}, such that this resolution X̃_{α}, smooth over S, contains X_{α} as a Zariski open; the complement X̃_{α}−X_{α} is also supposed to be a union of normal crossing divisors.

If X is an invariant subscheme of the affine Grassmannian or of the affine flag variety, we can use the Demazure resolution.

Let i_{α} denote the inclusion map X_{α} → X and let F_{α} denote i_{α,!}ℚ̄_{ℓ}. A bounded complex of sheaves F with constructible cohomology sheaves (more precisely an object of D_{c}^{b}(X, ℚ_{ℓ}) – cf. the second footnote), is said to be Δ-constant if the cohomology sheaves of F are successive extensions of F_{α} with α ∈ Δ. The intersection complex of X̄_{α} is Δ-constant.

For an ℓ-adic complex F of sheaves on X, there exists a canonical morphism

Fs̄ → RΨ^{X}(F_{η})

whose mapping cylinder is the vanishing cycle RΦ^{X}(F).

Lemma 16 If F is a Δ-constant complex, then RΦ^{X}(F) = 0.

Proof. Clearly, it suffices to prove RΦ^{X}(F_{α}) = 0. Consider the equivariant resolution π_{α} : X̃_{α} → X̄_{α}. We have a canonical isomorphism

Rπ_{α,∗}RΦX̃_{α}(F_{α}) ⟶^{∼} RΦX̄_{α}(F_{α}).

It suffices then to prove RΦX̃_{α}(F_{α}) = 0. This is known because X̃_{α} is smooth over S and X̃_{α} − X_{α} is a union of normal crossing divisors. □

Corollary 17 If F is Δ-constant and bounded, the inertia group Γ_{0} acts trivially on the nearby cycle RΨ^{X}(F_{η}).

Proof. The morphism Fs̄ → RΨ^{X}(F_{η}) is an isomorphism compatible with the actions of Γ. The inertia subgroup Γ_{0} acts trivially on Fs̄, thus it acts trivially on RΨ^{X}(F_{η}), too. □

6 Invariant subschemes of G/I

We recall here the well known ind-scheme structure of G_{k}/I_{k} where G denotes the group GL(d, k((t + ϖ))) or the group GSp(2d, k((t + ϖ))) and where I is its standard Iwahori subgroup. The variable t + ϖ is used instead of t in order to be compatible with the definitions of local models given in section 2.

6.1 Linear case

Let N_{n_{±}} be the functor which associates to each O-algebra R the set of

L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d} = (t + ϖ)^{−1}L_{0})

where L_{0}, L_{1}, … are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{d} such that for i = 0, 1, …, d − 1

(t + ϖ)^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ (t + ϖ)^{n_{−}}V_{i,R}

and L_{i}/(t + ϖ)^{n_{+}}V_{i,R} is locally a direct factor, of fixed rank independent of i, of the free R-module (t + ϖ)^{n_{−}}V_{i,R}/(t + ϖ)^{n_{+}}V_{i,R}. Obviously, this functor is represented by a closed subscheme in a disjoint union of products of Grassmannians. In particular, N_{n_{±}} is proper.

Let I_{n_{±}} be the functor which associates to each O-algebra R the group R[t]-linear automorphisms of

(t + ϖ)^{n_{−}−1}R[t]^{d}/(t + ϖ)^{n_{+}}R[t]^{d}

fixing the image in this quotient of the filtration

V_{0,R} ⊂ V_{1,R} ⊂ ⋯ ⊂ V_{d,R} = (t + ϖ)^{−1}V_{0,R}.

This functor is represented by a smooth group scheme over S which acts on N_{n_{±}}.

6.2 Symplectic case

Let N_{n_{±}} be the functor which associates to each O-algebra R the set of sequences

L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d})

where L_{0}, L_{1}, … are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{2d} satisfying

(t + ϖ)^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ (t + ϖ)^{n_{−}}V_{i,R}

and such that L_{i}/(t + ϖ)^{n_{+}}V_{i,R} is locally a direct factor of (t + ϖ)^{n_{−}}V_{i,R}/(t + ϖ)^{n_{+}}V_{i,R} of rank (n_{+} − n_{−})d for all i = 0, 1, …, d, and L_{0} (resp. L_{d}) is autodual with respect to the symplectic form (t + ϖ)^{−n_{−}−n_{+}}⟨ , ⟩ (resp. (t + ϖ)^{−n_{−}−n_{+}+1}⟨ , ⟩).

Let I_{n_{±}} be the functor which associates to each O-algebra R the group R[t]-linear automorphisms of

(t + ϖ)^{n_{−}−1}R[t]^{2d}/(t + ϖ)^{n_{+}}R[t]^{2d}

fixing the image in this quotient, of the filtration

V_{0,R} ⊂ V_{1,R} ⊂ ⋯ ⊂ V_{2d,R} = (t + ϖ)^{−1}V_{0,R},

and fixing the symplectic form (t + ϖ)^{−n_{−}−n_{+}+1}⟨ , ⟩ up to a unit in R. This functor is represented by a smooth group scheme over S which acts on N_{n_{±}}.

6.3 There is no vanishing cycle on N

For any algebraically closed field k over O, each N_{n_{±}}(k) is an I_{n_{±}}-invariant subset of the direct limit

\underrightarrow{lim}_{n_{±}→±∞} N_{n_{±}}(k) = G(k((t + ϖ)))/I_{n_{±}}

where G denotes either the linear group or the group of symplectic similitudes. It follows from the Bruhat-Tits decomposition that N_{n_{±}} admits a stratification by I_{n_{±}}-orbits

N_{n_{±}} = ∐_{w∈W̃′(n_{±})} O_{w}

where W̃′(n_{±}) is a finite subset of the affine Weyl group W̃ of GL(d) (resp. GSp(2d)). Moreover, for all w ∈ W̃′(n_{±}), O_{w} is isomorphic to the affine space 𝔸_{S}^{ℓ(w)} of dimension ℓ(w) over S, in particular it is smooth over S. By construction, I_{n_{±}} acts transitively on each O_{w}. All this remains true if we replace S by any other base scheme.

Let Ō_{w} denote the closure of O_{w}. Let I_{w,η} (resp. I_{w,s}) denote the intersection complex of Ō_{w,η} (resp. Ō_{w,s}). We have

RΨ^{N}(I_{w,η}) = I_{w,s̄}

(see Appendix 5.2 for a proof). In particular, the inertia subgroup Γ_{0} acts trivially on RΨ^{N}(I_{w,η}).

Let W̃ be the affine Weyl group of GL(d), respectively GSp(2d). It can be easily checked that W̃ = ⋃_{n_{±}} W̃′(n_{±}) for the linear case as well as for the symplectic case.

7 Convolution product of A_{λ} with I_{w}

7.1 Convolution diagram

In this section, we will adapt a construction due to Lusztig in order to define the convolution product of an equivariant perverse sheaf A_{λ} over M_{n_{±}} with an equivariant perverse sheaf I_{w} over N_{n′_{±}}. See Lusztig’s article [15] for a quite general construction.

For any dominant coweight λ and any w ∈ W̃, we can choose n_{±} and n′_{±} so that λ ∈ Λ(n_{±}) and w ∈ W̃′(n′_{±}). From now on, since λ and w as well as n_{±} and n′_{±} are fixed, we will often write M for M_{n_{±}} and N for N_{n′_{±}}. This should not cause any confusion.

The aim of this subsection is to construct the convolution diagram à la Lusztig

M̃ × Ñ
p_{1} ↙  ↘^{p_{2}}
M × N  M ˜× N ⟶^{m} P

with the usual properties that will be made precise later.

7.2 Linear case

• The functor M ˜× N associates to each O-algebra R the set of pairs (L_{•}, L′_{•})

L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d} = (t + ϖ)^{−1}L_{0})
L′_{•} = (L′_{0} ⊂ L′_{1} ⊂ ⋯ ⊂ L′_{d} = (t + ϖ)^{−1}L′_{0})

where L_{i}, L′_{i} are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{d} satisfying the following conditions

t^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ t^{n_{−}}V_{i,R}
(t + ϖ)^{n′_{+}}L_{i} ⊂ L′_{i} ⊂ (t + ϖ)^{n′_{−}}L_{i}

As usual, L_{i}/t^{n_{+}}V_{i,R} is supposed to be locally a direct factor of t^{n_{−}}V_{i,R}/t^{n_{+}}V_{i,R}, and L′_{i}/(t + ϖ)^{n′_{+}}L_{i} locally a direct factor of (t + ϖ)^{n′_{−}}L_{i}/(t + ϖ)^{n′_{+}}L_{i} as R-modules.The ranks of the projective R-modules L_{i}/t^{n_{+}}V_{i,R} and L′_{i}/(t + ϖ)^{n′_{+}}L_{i} are each also supposed to be independent of i. It follows from the above conditions that

t^{n_{+}}(t + ϖ)^{n′_{+}}V_{i,R} ⊂ L′_{i} ⊂ t^{n_{−}}(t + ϖ)^{n′_{−}}V_{i,R}

and L′_{i}/t^{n_{+}}(t+ϖ)^{n′_{+}}V_{i,R} is locally a direct factor of t^{n_{−}}(t+ϖ)^{n′_{−}}V_{i,R}/t^{n_{+}}(t+ ϖ)^{n′_{+}}V_{i,R} as an R-module. Thus defined the functor M ˜× N is represented by a projective scheme over S.

• The functor P associates to each O-algebra R the set of chains L′_{•}

L′_{•} = (L′_{0} ⊂ L′_{1} ⊂ ⋯ ⊂ L′_{d} = (t + ϖ)^{−1}L′_{0})

where L′_{i} are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{d} satisfying

t^{n_{+}}(t + ϖ)^{n′_{+}}V_{i,R} ⊂ L′_{i} ⊂ t^{n_{−}}(t + ϖ)^{n′_{−}}V_{i,R}

and the usual conditions “locally a direct factor as R-modules”. As above, rk_{R}(L′_{i}/t^{n_{+}}(t + ϖ)^{n′_{+}}V_{i,R}) is supposed to be independent of i. Obviously, this functor is represented by a projective scheme over S.

• The forgetting map m(L_{•}, L′_{•}) = L′_{•} yields a morphism

m : M ˜× N → P.

This map is defined: it suffices to note that t^{n_{−}}(t + ϖ)^{n′_{−}}V_{i,R}/L′_{i} is locally free as an R-module, being an extension of t^{n_{−}}V_{i,R}/L_{i} by (t + ϖ)^{n′_{−}}L_{i}/L′_{i}, each of which is locally free. Clearly, this morphism is a proper morphism because its source and its target are proper schemes over S.

Now before we can construct the schemes M̃, Ñ, and the remaining morphisms in the convolution diagram, we need the following simple remark.

Lemma 18 The functor which associates to each O-algebra R the set of matrices g ∈ 𝔤𝔩_{s}(R) such that the image of g : R^{s} → R^{s} is locally a direct factor of rank r of R^{s} is representable by a locally closed subscheme of 𝔤𝔩_{s}.

Proof. For 1 ≤ i ≤ s, denote by St_{i} the closed subscheme of 𝔤𝔩_{s} defined by the equations: all minors of order at least i + 1 vanish. By using Nakayama’s lemma, one can see easily that the above functor is represented by the quasiaffine, locally closed subscheme St_{r} − St_{r−1} of 𝔤𝔩_{s}. □

Now let V̄_{0} ⊂ V̄_{1} ⊂ ⋯ be the image of V_{0} ⊂ V_{1} ⊂ ⋯ in the quotient

V̄ = t^{n_{−}}(t + ϖ)^{n′_{−}−1}O[t]^{d}/t^{n_{+}}(t + ϖ)^{n′_{+}}O[t]^{d}.

Let L̄_{0} ⊂ L̄_{1} ⊂ ⋯ be the images of L_{0} ⊂ L_{1} ⊂ ⋯ in the quotient V̄_{R} = V̄ ⊗_{O}R. Because L_{i} is completely determined by L̄_{i}, we can write L̄_{•} ∈ M(R) for L_{•} ∈ M(R) and so on.

• We consider the functor M̃ which associates to each O-algebra R the set of R[t]-endomorphisms g ∈ End(V̄_{R}) such that if L̄_{i} = g(t^{n_{−}} V̄_{i}) then

t^{n_{+}} V̄_{i,R} ⊂ L̄_{i} ⊂ t^{n_{−}} V̄_{i,R}

and L̄_{i}/t^{n_{+}} V̄_{i,R} is locally a direct factor of t^{n_{−}} V̄_{i,R}/t^{n_{+}} V̄_{i,R}, of the same rank, for all i = 0, …, d − 1. Using Lemma 18 one sees this functor is representable and comes naturally with a morphism p : M̃ → M.

• In a totally analogous way, we consider the functor Ñ which associates to each O-algebra R the set of R[t]-endomorphisms g ∈ End(V̄_{R}) such that if L̄_{i} = g((t + ϖ)^{n′_{−}} V̄_{i,R}) then

(t + ϖ)^{n′_{+}} V̄_{i,R} ⊂ L̄_{i} ⊂ (t + ϖ)^{n′_{−}} V̄_{i,R}

and L̄_{i}/(t + ϖ)^{n′_{+}} V̄_{i,R} is locally a direct factor of (t + ϖ)^{n′_{−}} V̄_{i,R}/(t + ϖ)^{n′_{+}} V̄_{i,R}, of the same rank for all i = 0, …, d − 1. As above, the representability follows from Lemma 18. This functor comes naturally with a morphism p′ : Ñ → N.

• Now we define the morphism p_{1} : M̃ × Ñ → M × N by p_{1} = p × p′.

• We define the morphism p_{2} : M̃ × Ñ → M ˜× N by p_{2}(g, g′) = (L_{•}, L′_{•})

with

(L_{•}, L′_{•}) = (g(t^{n_{−}}V_{•}), gg′(t^{n_{−}}(t + ϖ)^{n′_{−}}V_{•})).

We have now achieved the construction of the convolution diagram. We need to prove some facts related to this diagram.

Lemma 19 The morphisms p_{1} and p_{2} are smooth and surjective. Their restrictions to connected components of M̃ ×Ñ with image in the corresponding connected components of M × N and of M ˜×N, have the same relative dimensions.

Proof. The proof is very similar to that of Lemma 3. Let us note that the morphism p : M̃ → M can be factored as p = f ∘ j where j : M̃ → U is an open immersion and f : U → M is the vector bundle defined as follows. For any O-algebra R and any L_{•} ∈ M(R), the fiber of U over L_{•} is the R-module

U(L_{•}) = ⨁_{i=0}^{d−1}(t + ϖ)^{n′_{−}}L_{i}/t^{n_{+}}(t + ϖ)^{n′_{+}}V_{i,R}.

The morphisms p′, p_{1} and p_{2} can be described in the same manner. The equality of relative dimensions of p_{1} and p_{2} follows from Lemma 24 (proved in section 8) and the fact that they are each smooth. □

Just as in subsection 2.2, we can consider the group valued functor J̃ which associates to each O-algebra R the group of R[t]-linear automorphisms of V̄_{R} which fix the filtration V̄_{0} ⊂ V̄_{1} ⊂ ⋯ ⊂ V̄_{d}. Obviously, this functor is represented by a connected affine algebraic group scheme over S. The same proof as that of Lemma 3 proves that J̃ is smooth over S. Moreover, there are canonical morphisms of S-group schemes J̃ → J and J̃ → I, where J = J_{n_{±}} (resp. I = I_{n′_{±}}) is the group scheme defined in subsection 2.2 (resp. 6.1).

• We consider the action α_{1} of J̃ × J̃ on M̃ × Ñ defined by

α_{1}(h, h′; g, g′) = (gh^{−1}, g′h′^{−1}).

Clearly, this action leaves stable the fibers of p_{1} : M̃ × Ñ → M × N.

• We also consider the action α_{2} of J̃× J̃ on the same M̃ × Ñ defined by

α_{2}(h, h′; g, g′) = (gh^{−1}, hg′h′^{−1}).

Clearly, this action leaves stable the fibers of p_{2} : M̃ × Ñ → M ˜× N.

Lemma 20 (i) The action α_{1}, respectively α_{2}, is transitive on all geometric fibers of p_{1}, respectively p_{2}. The geometric fibers of p_{1}, respectively p_{2}, are therefore connected.

(ii) Moreover, the stabilizer under the action α_{1}, respectively α_{2}, of any geometric point is a smooth connected subgroup of J̃ × J̃.

Proof. Let E be a (separably closed) field containing the fraction field F of O or its residue field k. Let g, g′ be elements of M̃(E) such that

L_{•} = p(g) = p(g′) ∈ M(E).

For all i = 0, …, d − 1, denote by V̂_{i} and L̂_{i} the tensors

V̂_{i} = V_{i} ⊗_{O[t]} E[t]_{(t(t+ϖ))}
L̂_{i} = L_{i} ⊗_{E[t]} E[t]_{(t(t+ϖ))}

where E[t]_{(t(t+ϖ))} is the localized ring of E[t] at the ideal (t(t + ϖ)), i.e., the ring S^{−1}E[t] where S = E[t] − {(t) ∪ (t + ϖ)}; this is a semi-local ring. Of course, we can consider the modules V̂_{i} and L̂_{i} as E[t]_{(t(t+ϖ))}-submodules of E(t)^{d}.

Clearly, we have an isomorphism

V̄_{E} = t^{n_{−}}(t + ϖ)^{n′_{−}−1}V̂_{0}/t^{n_{+}}(t + ϖ)^{n′_{+}} V̂_{0}

so that E[t]-endomorphisms of V̄_{E} are the same as E[t]_{(t(t+ϖ))}-endomorphisms of V̂_{0} taken modulo t^{n_{+}−n_{−}}(t + ϖ)^{n′_{+}−n′_{−}+1}.

By using the Nakayama lemma, g and g′ can be lifted to

ĝ, ĝ′ ∈ GL(d, E(t))

such that

L̂_{i} = ĝt^{n_{−}} V̂_{i} ; L̂_{i} = ĝ′t^{n_{−}} V̂_{i}.

This induces of course ĥV̄_{i} = V̄_{i} with ĥ = ĝ^{−1}ĝ′ and for all i = 0, …, d − 1.

Let h be the reduction modulo t^{n_{+}−n_{−}}(t+ϖ)^{n′_{+}−n′_{−}+1} of ĥ. It is clear that g′ = gh and h lies in J̃(E).

We have proved that J̃ acts transitively on the geometric fibers of M̃ → M. We can prove in a completely similar way that J̃ acts transitively on the geometric fibers of Ñ → N. Consequently, the action α_{1} is transitive on the geometric fibers of p_{1}.

The proof of the statement for α_{2} and p_{2} is similar. This completes the proof of (i).

For (ii), let L_{•} ∈ M(E) and take g ∈ M̃(E) over L_{•}. We have to look at the points h ∈ J̃(E) such that gh = g as endomorphisms of V̄_{E}. Let e_{1}, …, e_{d} be the standard generators of V̄_{E} so that the image in V̄_{E} of t^{n_{−}}(t + ϖ)^{n′_{−}}V_{i} is generated by e_{1}, …, e_{i}, (t + ϖ)e_{i+1}, …, (t + ϖ)e_{n}. Let us denote that image by t^{n_{−}}(t + ϖ)^{n′_{−}} V̄_{i}. Now gh = g if and only if h(e_{i}) − e_{i} belongs to Ker(g) for all i = 1, …, d. The condition h ∈ J̃(E) says that h(e_{i}) lies in the submodule generated by t^{n_{−}}(t + ϖ)^{n′_{−}} V̄_{i} and h is invertible. The dimension of the E-vector space Ker(g) ∩ t^{n_{−}}(t + ϖ)^{n′_{−}} V̄_{i} depends only on L_{•} and it is constant along each connected component of M as the dimension of g(t^{n_{−}}(t + ϖ)^{n′_{−}} V̄_{i}) = (t + ϖ)^{n′_{−}} L̄_{i} is. This proves that the stabilizer group scheme of J̃ acting on M̃ (i.e., the subscheme of J̃× M̃ on which the action and projection morphisms a, pr_{2} : J̃× M̃ → M̃ agree) is an open subscheme of a vector bundle over M̃. This shows that the stabilizer of a single point g ∈ M̃(E) is a connected smooth subgroup.

The same proof works for the actions α_{1} and α_{2}. □

We remark that Lemmas 19 and 20 are essential for the construction of the convolution of perverse sheaves, discussed in section 7.4.

The symmetric construction yields the following diagram

Ñ × M̃
p′_{1} ↙  ↘^{p′_{2}}
N × M  N ˜× M ⟶^{m′} P

enjoying the same structures and properties. More precisely, we define N ˜× M as follows: for each O-algebra R, let (N ˜× M)(R) be the set of pairs (L′_{•}, L_{•})

L′_{•} = (L′_{0} ⊂ L′_{1} ⊂ ⋯ ⊂ L′_{d} = (t + ϖ)^{−1}L′_{0})
L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d} = (t + ϖ)^{−1}L_{0})

where L′_{i}, L_{i} are R[t]-submodules of R[t, t^{−1}, (t+ϖ)^{−1}]^{d} satisfying the following conditions

(t + ϖ)^{n′_{+}}V_{i,R} ⊂ L′_{i} ⊂ (t + ϖ)^{n′_{−}}V_{i,R}
t^{n_{+}}L′_{i} ⊂ L_{i} ⊂ t^{n_{−}}L′_{i}

such that for each i = 0, …, d − 1, the R-module L′_{i}/(t + ϖ)^{n′_{+}}V_{i,R} is locally a direct factor of (t + ϖ)^{n′_{−}}V_{i,R}/(t + ϖ)^{n′_{+}}V_{i,R}, and the R-module L_{i}/t^{n_{+}}L′_{i} is locally a direct factor of t^{n_{−}}L′_{i}/t^{n_{+}}L′_{i}. It is also supposed that rk_{R}(L′_{i}/(t + ϖ)^{n′_{+}}V_{i,R}) and rk_{R}(L_{i}/t^{n_{+}}L′_{i}) are independent of i.

The morphisms p′_{1}, p′_{2}, and m′ are defined in the obvious way: p′_{1} = p′ ×p, m′(L′_{•}, L_{•}) = L_{•}, and p′_{2}(g′, g) = (g′(t + ϖ)^{n′_{−}}V_{i,R} , g′g(t^{n_{−}}(t + ϖ)^{n′_{−}})V_{i,R}).

7.3 Symplectic case

In this section we construct the symplectic analogue of the convolution diagram just discussed. In particular we need to define the schemes M ˜× N, M̃, Ñ, P, and the morphisms p_{1}, p_{2}, and m. Moreover we need to construct the smooth group scheme J̃ which acts on the whole convolution diagram. Once this is done, defining the symplectic analogies of the actions α_{1} and α_{2}, proving the symplectic analogies of Lemmas 19 and 20, and defining the symmetric construction are all straightforward tasks and will be left to the reader.

• The functor M ˜× N associates to each O-algebra R the set of pairs (L_{•}, L′_{•})

L_{•} = (L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{d})
L′_{•} = (L′_{0} ⊂ L′_{1} ⊂ ⋯ ⊂ L′_{d})

where L_{i}, L′_{i} are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{2d} satisfying the following conditions

t^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ t^{n_{−}}V_{i,R}
(t + ϖ)^{n′_{+}}L_{i} ⊂ L′_{i} ⊂ (t + ϖ)^{n′_{−}}L_{i}

satisfying the usual ”locally direct factors as R-modules” conditions: L_{i}/t^{n_{+}}V_{i,R} is locally a direct factor of t^{n_{−}}V_{i,R}/t^{n_{+}}V_{i,R} of rank (n_{+}−n_{−})d and L′_{i}/(t+ϖ)^{n′_{+}}L_{i} is locally a direct factor of (t+ϖ)^{n′_{−}}L_{i}/(t+ϖ)^{n′_{+}}L_{i} of rank (n′_{+} − n′_{−})d. Moreover we suppose L_{0}, L_{d}, L′_{0} and L′_{d} are autodual with respect to t^{−n_{−}−n_{+}}⟨ , ⟩, t^{−n_{−}−n_{+}}(t + ϖ)⟨ , ⟩, t^{−n_{−}−n_{+}}(t + ϖ)^{−n′_{−}−n′_{+}}⟨ , ⟩ and t^{−n_{−}−n_{+}}(t + ϖ)^{−n′_{−}−n′_{+}+1}⟨ , ⟩ respectively.

• The functor P associates to each O-algebra R the set of chains L′_{•}

L′_{•} = (L′_{0} ⊂ L′_{1} ⊂ ⋯ ⊂ L′_{d})

where L′_{i} are R[t]-submodules of R[t, t^{−1}, (t + ϖ)^{−1}]^{2d} satisfying

t^{n_{+}}(t + ϖ)^{n′_{+}}V_{i,R} ⊂ L′_{i} ⊂ t^{n_{−}}(t + ϖ)^{n′_{−}}V_{i,R},

such that the usual “locally a direct factor as R-modules of rank (n_{+} − n_{−} +n′_{+} −n′_{−})d” condition holds, and such that L′_{0} and L′_{d} are autodual with respect to t^{−n_{−}−n_{+}}(t+ϖ)^{−n′_{−}−n′_{+}}⟨ , ⟩ and t^{−n_{−}−n_{+}}(t+ϖ)^{−n′_{−}−n′_{+}+1}⟨ , ⟩ respectively.

• The forgetting map m(L_{•}, L′_{•}) = L′_{•} yields a morphism m : M ˜× N → P. Clearly, m is a proper morphism between proper S-schemes.

• We consider the functor M̃ which associates to each O-algebra R the set of R[t]-endomorphisms g of

V̄_{R} = t^{n_{−}}(t + ϖ)^{n′_{−}−1}V_{0,R}/t^{n_{+}}(t + ϖ)^{n′_{+}}V_{0,R}

satisfying

⟨gx, gy⟩ = c_{g}t^{n_{+}−n_{−}}⟨x, y⟩

for some c_{g} ∈ R^{×}, and such that if L̄_{i} = g(t^{n_{−}} V̄_{i}) for i = 0, …, d, then we have

t^{n_{+}} V̄_{i,R} ⊂ L̄_{i} ⊂ t^{n_{−}} V̄_{i,R},

and L̄_{i}/t^{n_{+}} V̄_{i,R} is locally a direct factor of t^{n_{−}} V̄_{i,R}/t^{n_{+}} V̄_{i,R} of rank (n_{+} − n_{−})d. If g ∈ M̃(R) then one sees using the definitions that automatically, L̄_{•} = gt^{n_{−}}V_{•,R} ∈ M(R). The functor M̃ is representable and comes naturally with a morphism p : M̃ → M.

• Next consider the functor Ñ which associates to each O-algebra R the set of R[t]-endomorphisms g of V̄_{R} satisfying

⟨gx, gy⟩ = c_{g}(t + ϖ)^{n′_{+}−n′_{−}}⟨x, y⟩

for some c_{g} ∈ R^{×} and such that if L̄′_{i} = g(t + ϖ)^{n′_{−}} V̄_{i,R} for i = 0, …, d then we have

(t + ϖ)^{n′_{+}} V̄_{i,R} ⊂ L̄′_{i} ⊂ (t + ϖ)^{n′_{−}} V̄_{i,R},

and L̄′_{i}/(t + ϖ)^{n′_{+}} V̄_{i,R} is locally a direct factor of (t + ϖ)^{n′_{−}} V̄_{i,R}/(t + ϖ)^{n′_{+}} V̄_{i,R} of rank (n′_{+} − n′_{−})d. From the definitions one sees that L̄′_{•} ∈ N(R). The functor Ñ is representable and comes with a morphism p′ : Ñ → N.

• We define p_{1} = p × p′. We define p_{2} : M̃ × Ñ → M ˜×N exactly as in the linear case.

• We let J̃ denote the functor which associates to any O-algebra R the group of R[t]-linear automorphisms of V̄_{R} which fix the form t^{−n_{−}−n_{+}}(t+ ϖ)^{−n′_{−}−n′_{+}+1}⟨ , ⟩ up to an element in R^{×} and which fix the filtration V̄_{i,R}. As in Lemma 3, the group scheme J̃ is smooth over S. There are canonical S-group scheme morphisms J̃ → J and J̃ → I, where J = J_{n_{±}} (resp. I = I_{n′_{±}}) was defined in subsection 2.5 (resp. 6.2).

7.4 Definition of the convolution product

Let us recall the standard definition of convolution product due to Lusztig [15] (see also [6] and [16]).

Let E be a field containing the fraction field F of O or its residue field k and let ϵ = Spec (E) → S be the corresponding morphism. For all S-schemes X, let X_{ϵ} denote the base change X ×_{S} ϵ.

Let A be a perverse sheaf over M_{ϵ} that is J_{ϵ}-equivariant. Let I be a perverse sheaf over N_{ϵ} that is I_{ϵ}-equivariant. Both I_{ϵ} and J_{ϵ} are quotients of J̃_{ϵ}, so we can say that A and I are J̃_{ϵ}-equivariant.

Since p_{1} is a smooth morphism, the pull-back p_{1}^{∗}(A⊠_{ϵ}I) is also perverse up to the shift by the relative dimension of p_{1}. A priori, this pull-back is only α_{1}- equivariant. As A and I are J̃_{ϵ}-equivariant, p_{1}^{∗}(A ⊠_{ϵ} I) is also α_{2}-equivariant. Since p_{2} is smooth and the action α_{2} is transitive on its geometric fibers, the perverse sheaf F = p_{1}^{∗}(A ⊠_{ϵ} I) is constant along the fibers of p_{2}. Moreover the stabilizers for α_{2} of geometric points are smooth and connected. Under these hypotheses there exists a perverse sheaf A _{ ˜⊠_{ϵ}}I, unique up to unique isomorphism, such that

p_{1}^{∗}(A ⊠_{ϵ} I) = p_{2}^{∗}(A _{ ˜⊠_{ϵ}}I).

The uniqueness follows from Proposition 4.2.5 of [1] which only requires that p_{2} is smooth and its geometric fibers are connected.

To prove the existence of the perverse sheaf A _{ ˜⊠_{ϵ}}I we need the transitive group action on the fiber of p_{2}, the fact that p_{2} is smooth and surjective, and the fact that the action α_{2} has smooth connected stabilizers. We make use of the following general lemma.

Lemma 21 Suppose π : X → Y is a smooth surjective morphism of Sschemes, and suppose G_{Y} is a smooth connected Y -group scheme which acts trivially on Y and on X such that the action on each geometric fiber of π is transitive. Assume further that the stabilizer in G_{Y} of any geometric point of X is a smooth connected subgroup. Then a G_{Y} -equivariant perverse sheaf F on X descends along π.

Proof. Assume temporarily that π possesses a section s. Then the action map and the section s give rise to a morphism a : G_{Y} → X, which is smooth and surjective with geometrically connected fibers. Using the equivariance it follows that a^{∗}π^{∗}s^{∗}F = a^{∗}F. Since a and π ∘ a are both smooth with geometrically connected fibers, this implies that s^{∗}F is perverse up to the shift by the relative dimension of π. Indeed, since the other perverse cohomologies of s^{∗}F are killed by a^{∗}π^{∗}, they must be zero since this functor is fully faithful, by [1] 4.2.5. By applying the same proposition for a^{∗} now, we obtain an isomorphism between perverse sheaves π^{∗}s^{∗}F = F.

In the general case there is an étale covering U_{i} → Y such that each π_{i} : X ×_{Y} U_{i} → U_{i} has a section. Using the group action of G_{Y} ×_{Y} U_{i}, the previous discussion shows that étale locally F descends along π. Using 4.2.5 of loc.cit. again to descend the gluing data and using Theorem 3.2.4 of loc.cit. to glue perverse sheaves, we see that F descends along π globally. □

By Lemmas 19 and 20, the morphism p_{2} satisfies the hypotheses on π in Lemma 21, with G_{Y} = (J̃_{ϵ} × J̃_{ϵ}) × (M ˜×N) acting via α_{2}. We have thus proved the existence of A _{ ˜⊠_{ϵ}}I. Note that no shift is needed because on each connected component, p_{1} and p_{2} have the same relative dimension by Lemma 19.

Now set

A ∗_{ϵ} I = Rm_{∗}(A _{ ˜⊠_{ϵ}}I).

By the symmetric construction, we can define the convolution product I ∗_{ϵ}A.

Let E be now the algebraic closure k̄ of the residual field k. We suppose that the perverse sheaves A and I are equipped with an action of Gal(F̄/F) compatible with the action of Gal(F̄/F) on the geometric special fiber through Gal(k̄/k). In practice, the inertia subgroup Γ_{0} acts trivially on I and non trivially on A. As the semi-simple trace provides a sheaf-function dictionary, we have :

τ_{A}^{ss} ∗ τ_{I}^{ss} = τ_{A ∗s̄ I}^{ss}
τ_{I}^{ss} ∗ τ_{A}^{ss} = τ_{I ∗s̄ A}^{ss}

where the convolution on the left hand is the ordinary convolution in the Hecke algebra H(G_{k}//I_{k}). For the convenience of the reader let us recall the argument for this rather standard statement.

Recall that the convolution product f ∗ f′ of two functions in H(G_{k}//I_{k}) can be defined by

(f ∗ f′)(x) = ∑_{y∈G_{k}/I_{k}} f(y)f′(y^{−1}x)

for all x ∈ G_{k}/I_{k}. The right hand side is well defined since f′ is bi-I_{k}- invariant. If f = τ_{A}^{ss} and f′ = τ_{I}^{ss}, one can check the above sum is exactly the summation along the fiber of m of the semi-simple trace function associated to the perverse sheaf A _{ ˜⊠_{Spec(k)}}I. The compatibility between the ordinary convolution and the geometric convolution now follows from this, since the semi-simple trace behaves well with respect to a proper push-forward, as was made explicit in section 3.

8 Proof of Proposition 13

8.1 Cohomological part

According to the sheaf-function dictionary for semi-simple traces, it suffices to prove the following statement. Beilinson and Gaitsgory have proved a related result in the equal characteristic case, using a deformation of the affine Grassmannian of G, see [5].

Proposition 22 We have an isomorphism

RΨ^{M}(A_{λ,η}) ∗s̄ I_{w,s̄} ⟶^{∼} I_{w,s̄} ∗s̄ RΨ^{M}(A_{λ,η}).

Proof. The above statement makes sense because the functor RΨ sends perverse sheaves to perverse sheaves, by a theorem of Gabber, see Corollary 4.5 in [11]. In particular, RΨ^{M}(A_{λ,η}) is a perverse sheaf.

Let us recall that

RΨ^{N}(I_{w,η}) ⟶^{∼} I_{w,s}

so that we have to prove

RΨ^{M}(A_{λ,η}) ∗s̄ RΨ^{N}(I_{w,η}) ⟶^{∼} RΨ^{N}(I_{w,η}) ∗s̄ RΨ^{M}(A_{λ,η}).

First, let us prove that nearby cycle commutes with convolution product.

Lemma 23 We have the isomorphisms

RΨ^{M}(A_{λ,η}) ∗s̄ RΨ^{N}(I_{w,η}) ⟶^{∼} RΨ^{P} (A_{λ,η} ∗_{η} I_{w,η})

RΨ^{N}(I_{w,η}) ∗s̄ RΨ^{M}(A_{λ,η}) ⟶^{∼} RΨ^{P} (I_{w,η} ∗_{η} A_{λ,η})

Proof. According to a theorem of Beilinson-Bernstein (see Theorem 4.7 in [11]) we have an isomorphism of perverse sheaves

RΨ^{M×N}(A_{λ,η} ⊠_{η} I_{w,η}) ⟶^{∼} RΨ^{M}(A_{λ,η}) ⊠s̄ RΨ^{N}(I_{w,η}).

This induces an isomorphism between the pull-backs

p_{1}^{∗}RΨ^{M×N}(A_{λ,η} ⊠_{η} I_{w,η}) ⟶^{∼} p_{1}^{∗}(RΨ^{M}(A_{λ,η}) ⊠s̄ RΨ^{N}(I_{w,η}))

which are up to the shift by the relative dimension p_{1}, perverse too. By definition, we have

p_{1}^{∗}(RΨ^{M}(A_{λ,η}) ⊠s̄ RΨ^{N}(I_{w,η})) ⟶^{∼} p_{2}^{∗}(RΨ^{M}(A_{λ,η}) _{ ˜⊠s̄} RΨ^{N}(I_{w,η})).

As p_{1}, p_{2} are smooth, p_{1}^{∗} and p_{2}^{∗} commute with nearby cycle, so applying RΨ^{M̃×Ñ} to

p_{1}^{∗}(A_{λ,η} ⊠_{η} I_{w,η}) ⟶^{∼} p_{2}^{∗}(A_{λ,η}_{ ˜⊠_{η}}I_{w,η})

gives an isomorphism

p_{1}^{∗}RΨ^{M×N}(A_{λ,η} ⊠_{η} I_{w,η}) ⟶^{∼} p_{2}^{∗}RΨ^{M ˜× N}(A_{λ,η} _{ ˜⊠_{η}} I_{w,η}).

Since p_{2} is smooth with connected geometric fibers, Proposition 4.2.5 of Beilinson-Bernstein-Deligne [1] implies that we have an isomorphism

RΨ^{M ˜× N}(A_{λ,η} _{ ˜⊠_{η}} I_{w,η}) ⟶^{∼} RΨ^{M}(A_{λ,η}) _{ ˜⊠s̄} RΨ^{N}(I_{w,η}).

By applying now the functor Rm_{∗}, we have an isomorphism

Rm_{∗}RΨ^{M ˜× N}(A_{λ,η} _{ ˜⊠_{η}} I_{w,η}) ⟶^{∼} RΨ^{M}(A_{λ,η}) ∗s̄ RΨ^{N}(I_{w,η}).

Since the functor RΨ commutes with the direct image of a proper morphism, we have

RΨ^{P} (A_{λ,η} ∗_{η} I_{w,η}) ⟶^{∼} Rm_{∗}RΨ^{M ˜× N}(A_{λ,η} _{ ˜⊠_{η}} I_{w,η}).

By composing the above isomorphisms, we get

RΨ^{M}(A_{λ,η}) ∗s̄ RΨ^{N}(I_{w,η}) ⟶^{∼} RΨ^{P} (A_{λ,η} ∗_{η} I_{w,η}).

By the same argument, we prove

RΨ^{N}(I_{w,η}) ∗s̄ RΨ^{M}(A_{λ,η}) ⟶^{∼} RΨ^{P} (I_{w,η} ∗_{η} A_{λ,η}).

This finishes the proof of the lemma. □

Now it clearly suffices to prove

A_{λ,η} ∗_{η} I_{w,η} ⟶^{∼} I_{w,η} ∗_{η} A_{λ,η}

which is an easy consequence of the following lemma.

Lemma 24 1. Over the generic point η, we have two commutative triangles

M_{η} ˜× N_{η}
i ↙  ↘^{m}
M_{η} × N_{η} ⟶^{j} P_{η}
i′ ↖  ↗_{m′}
N_{η} ˜× M_{η}

where all arrows are isomorphisms.

2. Moreover, we have the following isomorphisms

i^{∗}(A_{λ,η} ⊠ I_{w,η}) ⟶^{∼} A_{λ,η} _{ ˜⊠} I_{w,η}

i′^{∗}(A_{λ,η} ⊠ I_{w,η}) ⟶^{∼} I_{w,η} _{ ˜⊠} A_{λ,η}

8.2 Proof of Lemma 24

Let us prove the above lemma in the linear case.

Over the generic point η, we have the canonical decomposition of

V̄_{F} = t^{n_{−}}(t + ϖ)^{n′_{−}−1}F [t]^{d}/t^{n_{+}}(t + ϖ)^{n′_{+}}F [t]^{d}

into the direct sum V̄_{F} = V̄_{F}^{(t)} ⊕ V̄_{F}^{(t+ϖ)} where

V̄_{F}^{(t)} = t^{n_{−}}F [t]^{d}/t^{n_{+}}F [t]^{d}
V̄_{F}^{(t+ϖ)} = (t + ϖ)^{n′_{−}−1}F [t]^{d}/(t + ϖ)^{n′_{+}}F [t]^{d}.

With respect to this decomposition, all the terms of the filtration

V̄_{0} ⊂ V̄_{1} ⊂ ⋯ ⊂ V̄_{d−1}

decompose to V̄_{i} = V̄_{i}^{(t)} ⊕ V̄_{i}^{(t+ϖ)} for all i = 0, …, d − 1. Here, we have

V̄_{0}^{(t)} = ⋯ = V̄_{d−1}^{(t)} = F [t]^{d}/t^{n_{+}}F [t]^{d}.

Let R be an F -algebra and let (L_{•}, L′_{•}) be an element of (M ˜× N)(R). These chains of R[t]-modules verify

t^{n_{+}}V_{i,R} ⊂ L_{i} ⊂ t^{n_{−}}V_{i,R}
(t + ϖ)^{n′_{+}}L_{i} ⊂ L′_{i} ⊂ (t + ϖ)^{n′_{−}}L_{i}

As usual, let L̄_{i}, L̄′_{i} denote the image of L_{i}, L′_{i} in V̄_{R}. As R[t]-modules, they decompose to L̄_{i} = L̄_{i}^{(t)}⊕L̄_{i}^{(t+ϖ)} and L̄′_{i} = L̄′_{i}^{(t)}⊕L̄′_{i}^{(t+ϖ)}. The above inclusion conditions imply indeed

L̄_{i}^{(t)} = L̄′_{i}^{(t)} ; L̄_{i}^{(t+ϖ)} = V̄_{i,R}^{(t+ϖ)}.

Consequently, L_{•} is completely determined by L′_{•}. In other terms, the map m(L̄_{•}, L̄′_{•}) = L̄′_{•} is an isomorphism of functors over η. In the same way, the map

i(L̄_{•}^{(t)} ⊕ V̄_{•,R}^{(t+ϖ)}, L̄_{•}^{(t)} ⊕ L̄′_{•}^{(t+ϖ)}) = (L̄_{•}^{(t)} ⊕ V̄_{•,R}^{(t+ϖ)}, V̄_{•,R}^{(t)} ⊕ L̄′_{•}^{(t+ϖ)})

yields an isomorphism i : M_{η} ˜× N_{η} ⟶^{∼} M_{η}×N_{η}. The composed isomorphism j = m ∘ i^{−1} is given by

j(L̄_{•}^{(t)} ⊕ V̄_{•,R}^{(t+ϖ)}, V̄_{•,R}^{(t)} ⊕ L̄′_{•}^{(t+ϖ)}) = L̄_{•}^{(t)} ⊕ L̄′_{•}^{(t+ϖ)}.

The analogous statement for the lower triangle in the diagram can be proved in the same way and the first part of the lemma is proved.

By the very definition of A_{λ,η} _{ ˜⊠} I_{w,η}, in order to prove the second part of the lemma, it suffices to construct an isomorphism

p_{1}^{∗}(A_{λ,η} ⊠ I_{w,η}) ⟶^{∼} p_{2}^{∗} i^{∗}(A_{λ,η} ⊠ I_{w,η}).

In fact, the triangle

M̃_{η} × Ñ_{η}
p_{1} ↙  ↘^{p_{2}}
M_{η} × N_{η} ⟵^{i} M_{η} ˜× N_{η}

does not commute. Nevertheless this lack of commutativity can be corrected by equivariance properties. We consider the diagram

M̃_{η} × Ñ_{η}
q_{1} ↙  ↘^{q_{2}}
J̃_{η} × M_{η} × N_{η} ⟵^{Id×i} J̃_{η} × M_{η} ˜× N_{η}
pr^{1} ↙  ↙_{pr_{2}}  ↘^{α}
M_{η} × N_{η} ⟵^{i} M_{η} ˜× N_{η}  M_{η} ˜× N_{η}

defined as follows.

For any F -algebra R, an element g ∈ M̃(R) is an R[t]-endomorphism of V̄_{R} such that L̄_{•} = g(t^{n_{−}} V̄_{•,R}) ∈ M(R). As V̄_{R} decomposes to V̄_{R} = V̄_{R}^{(t)} ⊕ V̄_{R}^{(t+ϖ)}, its R[t]-endomorphism g can be identified to a pair g = (g^{(t)}, g^{(t+ϖ)}) where g^{(t)}, respectively g^{(t+ϖ)}, is an endomorphism of V̄_{R}^{(t)}, respectively of V̄_{R}^{(t+ϖ)}.

As we have seen above, for L̄_{•} ∈ M(R), we have L̄_{i} = L̄_{i}^{(t)} ⊕ L̄_{i}^{(t+ϖ)} with L̄_{i}^{(t+ϖ)} = V̄_{i,R}^{(t+ϖ)}. Consequently, g^{(t+ϖ)} is an automorphism of V̄_{R}^{(t+ϖ)} fixing the filtration V̄_{•,R}^{(t+ϖ)}. In a similar way, an element g′ ∈ Ñ(R) can be identified with a pair (g′^{(t)}, g′^{(t+ϖ)}) where g′^{(t)} is an automorphism of V̄_{R}^{(t)} fixing the filtration V̄_{•,R}^{(t)}.

• The morphism q_{1} is defined by

q_{1}(g, g′) = ((g′^{(t)}, g^{(t+ϖ)}), g^{(t)}t^{n_{−}} V̄_{•,R}^{(t)}⊕V̄_{•,R}^{(t+ϖ)}, V̄_{•,R}^{(t)}⊕g′^{(t+ϖ)}(t+ϖ)^{n′_{−}} V̄_{•,R}^{(t+ϖ)}).

• The morphism q_{2} is defined by

q_{2}(g, g′) = ((g′^{(t)}, g^{(t+ϖ)}), g^{(t)}t^{n_{−}} V̄_{•,R}^{(t)}⊕V̄_{•,R}^{(t+ϖ)}, g^{(t)}t^{n_{−}} V̄_{•,R}^{(t)}⊕g′^{(t+ϖ)}(t+ϖ)^{n′_{−}} V̄_{•,R}^{(t+ϖ)}).

• The morphism α is defined by

α((g′^{(t)}, g^{(t+ϖ)}), L̄_{•}^{(t)} ⊕ V̄_{•,R}^{(t+ϖ)}, L̄_{•}^{(t)} ⊕ L̄′_{•}^{(t+ϖ)})
= (L̄_{•}^{(t)} ⊕ V̄_{•,R}^{(t+ϖ)}, L̄_{•}^{(t)} ⊕ g^{(t+ϖ)}L̄′_{•}^{(t+ϖ)}).

• pr_{1} and pr_{2} are the obvious projections

We can easily check that this diagram commutes and that

pr_{1} ∘ q_{1} = p_{1} ; α ∘ q_{2} = p_{2}.

Now it is clear that

p_{1}^{∗}(A_{λ,η} ⊠ I_{w,η}) ⟶^{∼} q_{2}^{∗} pr_{2}^{∗} i^{∗}(A_{λ,η} ⊠ I_{w,η}).

Moreover, by equivariant properties of A_{λ} and I_{w}, we have

pr_{2}^{∗} i^{∗}(A_{λ,η} ⊠ I_{w,η}) ⟶^{∼} α^{∗}i^{∗}(A_{λ,η} ⊠ I_{w,η}).

(Note that the group I_{η} acts on M_{η} ˜×N_{η} by acting on the second factor of M_{η} × N_{η} ≅ M_{η} ˜×N_{η} and α gives the corresponding action of J̃_{η} via the projection J̃_{η} → I_{η}.) In putting these things together, we get the required isomorphism

p_{1}^{∗}(A_{λ,η} ⊠ I_{w,η}) ⟶^{∼} p_{2}^{∗} i^{∗}(A_{λ,η} ⊠ I_{w,η}).

This finishes the proof of the lemma in the linear case.

In the symplectic case, let us mention that the F -vector space

t^{n_{−}}(t + ϖ)^{n′_{−}−1}F [t]^{2d}/t^{n_{+}}(t + ϖ)^{n′_{+}}F [t]^{2d}

equipped with the symplectic form t^{−n_{−}−n_{+}}(t + ϖ)^{−n′_{−}−n′_{+}+1}⟨ , ⟩ splits into the direct sum of two vector spaces

t^{n_{−}}F [t]^{2d}/t^{n_{+}}F [t]^{2d} ⊕ (t + ϖ)^{n′_{−}−1}F [t]^{2d}/(t + ϖ)^{n′_{+}}F [t]^{2d}

equipped with symplectic forms t^{−n_{−}−n_{+}}⟨ , ⟩ and (t+ϖ)^{−n′_{−}−n′_{+}+1}⟨ , ⟩ respectively. Further, note that g ∈ M̃(R) decomposes as g = (g^{(t)}, g^{(t+ϖ)}) where g^{(t)} ∈ Aut_{R[t]}(t^{n_{−}}R[t]^{2d}/t^{n_{+}}R[t]^{2d}) is such that ⟨g^{(t)}x, g^{(t)}y⟩ = c_{g^{(t)}}t^{−n_{−}+n_{+}}⟨x, y⟩ (for some c_{g^{(t)}} ∈ R^{×}), and g^{(t+ϖ)} ∈ Aut_{R[t]}((t+ϖ)^{n′_{−}−1}R[t]^{2d}/(t+ϖ)^{n′_{+}}R[t]^{2d}) is such that ⟨g^{(t+ϖ)}x, g^{(t+ϖ)}y⟩ = c_{g^{(t+ϖ)}}⟨x, y⟩ (for some c_{g^{(t+ϖ)}} ∈ R^{×}). A similar decomposition g′ = (g′^{(t)}, g′^{(t+ϖ)}) holds, and thus ones sees (g′^{(t)}, g^{(t+ϖ)}) ∈ J̃(R). Thus the maps q_{1} and q_{2} as defined above make sense in the symplectic case as well. The rest of the argument goes through without change as in the linear case.

This finishes the proof of Lemma 24. We have therefore finished the proof of Proposition 22, and thus Proposition 13 and Theorem 11 as well. □

9 The parahoric case

Similar results in the parahoric cases follow easily from the Iwahori case treated above.

Let G denote a split connected reductive over F, let K denote a special good maximal compact subgroup of G(F), let I denote an Iwahori subgroup contained in K, and let P denote a parahoric subgroup with I ⊂ P ⊂ K. We have the corresponding Hecke algebras of ℚ̄_{ℓ}-valued compactly supported bi-invariant functions H_{I}, H_{P}, and H_{K}. Let us define the convolution on the Hecke algebras using the Haar measures such that I (resp. P, K) has measure 1. There is a map between their centers

Z(H_{I}) → Z(H_{P})

given by z ↦ z^{P} = z ∗ 𝕀_{P}. (This is an algebra isomorphism; see the Remark following Theorem 25.)

Let G be one of the groups GL(d) or GSp(2d), and return to the notation G_{k}, I_{k}, H(G_{k}//I_{k}), etc. of section 4.2.

Assume we are in the linear case (the symplectic case is similar and will be omitted from this discussion). To a standard parahoric subgroup P_{k} with I_{k} ⊂ P_{k} ⊂ K_{k} we can associate a set of integers {0 = i_{0} < i_{1} < ⋯ < i_{p−1} < i_{p} = d} such that P_{k} is the stabilizer in G(k[[t]]) of the “standard” partial lattice chain V_{•}^{P} = (V_{0} ⊂ V_{i_{1}} ⊂ ⋯ ⊂ V_{i_{p}} = (t + ϖ)^{−1}V_{0}).

It is easy to define the analogue M_{r,n_{±}}^{P} of the local model M_{r,n_{±}} (cf. Definitions 1 and 2) as a scheme whose points are lattice chains

L_{•}^{P} = (L_{0} ⊂ L_{i_{1}} ⊂ ⋯ ⊂ L_{i_{p}} = (t + ϖ)^{−1}L_{0})

such that for every j, the lattice L_{i_{j}} is in a specified position relative to the lattices t^{n_{±}}V_{i_{j}}.

The obvious forgetful functor defines a proper map M_{r,n_{±}} → M_{r,n_{±}}^{P} which is an isomorphism over the generic fibers. By using the same argument as in section 5.1 we obtain the following result.

Theorem 25 Let λ be a dominant coweight of G = GL(d) or GSp(2d), and let P be a standard parahoric subgroup of G. Then

Tr^{ss}(Fr_{q}, RΨ^{M^{P}} (A_{λ,η})) = (−1)^{2⟨ρ,λ⟩} ∑_{λ′≤λ} m_{λ}(λ′)z_{λ′}^{P}.

Remark. Let W_{P} denote the parabolic subgroup of the Weyl group corresponding to P, and let W^{P} denote the set of minimal representatives for the cosets in W_{P} \W. One can show using the theory of the Bernstein center that we have the following commutative diagram of algebra isomorphisms

ℚ̄_{ℓ}[X_{∗}]^{W}

(−∗𝕀_{P})∘Bern. ↙  ↖^{Sat.}
Z(H(G_{k}//P_{k})) ⟶_{−∗𝕀_{I_{k}W^{P} I_{k}}} H(G_{k}//K_{k})

Therefore the right hand side in Theorem 25 can be characterized as follows: it is the unique element in Z(H(G_{k}//P_{k})) such that the Satake transform of its image under −∗𝕀_{I_{k}W^{P} I_{k}} is equal to (−1)^{2⟨ρ,λ⟩}χ_{λ}.

References

[1] A.A. Beilinson ; J. Bernstein ; P. Deligne. Faisceaux pervers in Analyse et topologie sur les espaces singuliers, I Astérisque 100 (1982).

[2] J.-F. Dat. Caractères à valeurs dans le centre de Bernstein. J. Reine Angew. Math. 508 (1999), 61–83.

[3] P. Deligne. Le formalisme des cycles évanescents in SGA 7 II, LNM 340, Springer 1973.

[4] P. Deligne. La conjecture de Weil II. Inst. Hautes Études Sci. Publ. Math. No. 52 (1980), 137-252.

[5] D. Gaitsgory. Construction of central elements in the affine Hecke algebra via nearby cycles. Inv. Math. 144 (2001) 253-280.

[6] V. Ginzburg. Perverse sheaves on a loop group and Langlands duality. Preprint (1996)

[7] A. Grothendieck. Formule de Lefschetz et rationalité des fonctions L Sém. Bourbaki no 279.

[8] T. Haines. The combinatorics of Bernstein functions. Trans. A.M.S. 353 (2001), 1251-1278.

[9] T. Haines. Test Functions for Shimura Varieties: the Drinfeld Case. Duke Math. J. 106 (2001), 19-40.

[10] T. Haines, B.C. Ngô. Alcoves associated to special fibers of local models. preprint (2000), math.RT/0103048 (to appear Amer. J. Math.).

[11] L. Illusie. Autour du théorème de monodromie locale in Périodes padiques. Astérisque No. 223 (1994), 9–57.

[12] S.-I. Kato. Spherical functions and a q-analogue of Kostant’s weight multiplicity formula. Invent. Math. 66 (1982), no. 3, 461–468.

[13] R. Kottwitz, M. Rapoport. Minuscule Alcoves for Gl_{n} and GSp_{2n}. Manuscripta Math. 102 (2000) 403-428.

[14] G. Lusztig. Singularities, characters formulas and a q-analogue of weight multiplicities in Analyse et topologie sur les espaces singuliers. Astérisque 101-102 (1983), 200-229.

[15] G. Lusztig. Cells in affine Weyl groups and tensor categories. Advances in Math. 129(1997), 85–98.

[16] I. Mirkovic, K. Vilonen. Perverse sheaves on loop grassmannians and Langlands duality.Math. Res. Lett. 7, No.1, 13-24 (2000).

[17] M. Rapoport. On the bad reduction of the Shimura varieties in Automorphic forms, Shimura varieties and L-functions edited by L.Clozel and J. Milne. Perspec. Math. 11, p. 253-321, Acad. Press 1990.

[18] M. Rapoport,Th. Zink. Period spaces for p-divisible groups. Annals of Math. Studies 144, Princeton Univ. Press 1996

[19] I. Satake. Theory of spherical functions on reductive algebraic groups over p-adic fields. Publ. IHES, 18: 1–69, 1963.

University of Toronto Department of Mathematics 100 St. George Street Toronto, ON M5S 3G3, Canada email:haines@math.toronto.edu

CNRS, Université de Paris Nord Département de mathématiques, 93430 Villetaneuse, FRANCE email: ngo@math.univ-paris13.fr