TOWARDS A GEOMETRIC CORRESPONDENCE FOR σ-DERIVATIONS

by Marc Reversat

Abstract. — This paper is a review of a tentative to give a common formalism for an analog of the geometric Satake correspondance, when the usual derivation is replaced by a σ-derivation, for essentially two kinds of morphisms σ: on one hand σ is a Frobenius morphism and then we work with schemes, bundles… over a base field of positive characteristic, on the other hand σ is a q-difference morphism and we are over ℂ. First we explain the motivation of the problem, second we give the main ideas of our construction and sketches of proofs.

Résumé (Vers une correspondance géométrique pour les σ-dérivations)

Nous expliquons ici une tentative de construction d’une sorte de correspondance géométrique de Satake, quand la dérivation usuelle est remplacée par des σ-dérivations, ceci dans deux “cas extrèmes” : lorsque σ est un morphisme de Frobenius, alors nous travaillons avec des schémas, fibrés… sur un corps de base de caractéristique positive, ou bien lorque σ est un morphisme aux q-différences, le corps de base étant cette fois ℂ. D’abord nous expliquons les motivations d’ un tel problŔme, ensuite nous donnons les principaux arguments de notre construction ainsi qu’une idée des démonstrations.

Preleminary version, february 24, 2003. Marc Reversat, Laboratoire de Mathématiques Émile Picard, Unité Mixte de Recherches Université/C.N.R.S. 5580, Université Paul Sabatier-Toulouse 3, 118, route de Narbonne, 31062 Toulouse cédex 4, France.

E-mail : reversat@picard.ups-tlse.fr
Url : http://picard.ups-tlse.fr/~reversat/

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. The affine grassmannian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. Topology and sheaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4. Derivations and differentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5. D_{σ}-modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6. A tannakian category. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7. The main result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1. Introduction

During the seventies has appeared clearly analogies between Drinfeld modules, a notion coming from the arithmetic in positive characteristics, and a class of differential equations (of course living in spaces of zero characteristic), the Krichever modules.

Let C be a smooth projective and geometrically irreducible curve over a field 𝔽_{q} with q elements (resp. over the field ℂ of complex numbers), let ∞ be a close point of C and A = O_{C}(C − {∞}) be the ring of regular functions outside ∞. Let k be an algebra over 𝔽_{q} (resp. be a differential algebra over ℂ). When the base field is 𝔽_{q}, we denote by L = k{τ} the ring of Ore polynomials (cf [34]), where τ is the Frobenius morphism such that for all λ in k, τλ = λ^{q}τ. When ℂ is the base field, we set L = k[D], where D is a differential operator such that for all λ in k, Dλ − λD is the derivative of λ. Let

u = ∑_{0≤i≤d} u_{i}τ^{i} , resp. u = ∑_{0≤i≤d} u_{i}D^{i}

be an element of L, the order of u is d if u_{d} ≠ 0 and u is called elliptic if u_{d} is invertible in k. A morphism of algebras

ϕ : A → L (1)

is called a Drinfeld module (resp. a Krichever module) if ϕ(a) is elliptic for all non zero element of A and if there exists an a ∈ A such that ϕ(a) has non zero order.

σ-DERIVATIONS 3

Drinfeld modules has been introduced, under the name of elliptic modules, by V.G.Drinfeld ([11]), to construct arithmetic objects of positive characteristic analog to those coming from the elliptic curves over the field of rationals ℚ, as (non commutative) rings of isogenies and their orders, torsion points and Tate modules, modular schemes…, which permit to Drinfeld to give a proof of the Langlands correspondence in positive characteristic for GL_{2} and for Galois representations that “look like the special representation at infinity”(op. cit. and [13]).

Commutative subrings of L = k[D], especially when k = ℂ[[t]] and D = d/dt, has been introduced by I.M. Krichever to study differential equations, he proved that for any commutative subring B of L = ℂ[[t]][d/dt] containing two operators with coprime orders, there exists a curve C over ℂ and a morphism as in (1), with image equal to B ([19], see also [20] ); this means that B is what is called today a Krichever module.

V.G. Drinfeld has given the common framework for this two kind of modules: in both case (positive characteristic case and differential case), the commutative subrings of L are of the form (1), this is proved by introducing a common formalism ([13]). At almost the same time, D. Mumford ([33]) has given dictionaries relating in three different cases, on one hand an algebraic curve, one or many of its points and a vector bundle on this curve, on the other hand commutative subrings of rings of operators. Firstly D. Mumford considers the case of a difference operator (for instance, with the previous notation L = k(t){σ}, where σ is defined by σ(f(t)) = f(t+1)), in the second case D. Mumford recovers Krichever results recalled above, the third ditionary extends (at least partially) the Drinfeld framework (see above) to the case where L = k{σ}, k beeing any field and σ ∈ Aut(k) beeing of infinite order. Note that the three situations studied here can be decribed by derivations, ordinary derivation in the Krichever case, σ-derivation^{(1)} in the other cases, the derivation over k beeing δ = σ − Id_{k}.

Modular schemes exist for Drinfeld modules ([11], indeed they have been generalized, see [29] and [22]) and for Krichever modules ([24], unfortunately unpublished). In a forthcoming paper ([36]) we will explain how to construct modular schemes for σ-difference equations, at least in the q-difference case (this construction uses arguments closed to those of [11] and [24]). But the aim of this paper is slitly different, it is in the spirit of works [12] of Drinfeld and [33] of Mumford: to find a common formalism for phenomenons occuring for σ-derivations, when σ is the identity map (the ordinary derivation case), or when σ is a Frobenius morphism (a situation closed to the Drinfeld one) and in other cases, especially when σ is a q-difference morphism (let q ∈ k, q ≠ 0, 1, this is for instance the case where L = k(t){σ} with σ defined by σ(f(t)) = f(qt)). Because our problem comes from the Langlands program, we recall now some of its aspects (indeed a few of them!).

(1)Definition is given in § 4.1.

Let C/𝔽_{q} be a projective, smooth and geometrically irreducible curve over the finite field 𝔽_{q} and let F be its field of rational functions (i.e. F is a global field of positive characteristic). V.G. Drinfeld has proved for GL_{2}(F) the so-called Langlands correspondence (see [13]), which has been completely generalized to GL_{n}(F) by L. Lafforgue ([23]), but we are here rather interested by an other aspect of the Langlands philosophy, which appear clearly in [14]. Let l be a prime number, l ≠ 0 in 𝔽_{q}. In [14], Drinfeld has proved that there exists a natural correspondence between 2-dimensional continuous l- adic representations of π_{1}(C) and the set of unramified cuspidal Hecke eigenforms. We give some details for this statement, because it is important for the sequel of this introduction.

Let F^{alg} be an algebraic closure of F (the field of rational functions over C) and 𝔸_{F} de the adèle ring of F. For any closed point ν of C we denote by F_{ν} the completion of F at the place ν, by O_{ν} the valuation ring of F_{ν}, by q_{ν} the cardinality of its residue field and by σ_{ν} ∈ π_{1}(C) = Gal(F^{alg}/F) a lift of the Frobenius morphism at ν. An unramified cusp form is a function f : GL_{2}(𝔸_{F}) → ℚ_{l}^{alg} (an algebraic closure of ℚ_{l}) which is right invariant under GL_{2}(F) and left invariant under GL_{2}(𝕆) (where 𝕆 is the product of all O_{ν}), satifying the cusp condition: for all g ∈ GL_{2}(𝔸_{F})

∫_{h∈𝔸_{F} /F} f(g (\begin{matrix}1 & h \\ 0 & 1 \end{matrix}))dh = 0.

For every closed point ν of C let T_{ν} and U_{ν} be the two Hecke operators defined on the space of cusp forms by

T_{ν}(f)(g) = ∫_{M_{ν}} f(h^{−1}g)dh and U_{ν}(f)(g) = f((\begin{matrix}ϖ_{ν}^{−1} & 0 \\ 0 & ϖ_{ν}^{−1} \end{matrix}) g),

where ϖ_{ν} is a uniformizing parameter in O_{ν}, where M_{ν} is the set of 2×2 matrices with entries in O_{ν} and determinant in ϖ_{ν}O_{ν}^{∗}, and where dh is here the Haar measure on GL_{2}(F_{ν}) normalized by mes(GL_{2}(O_{ν})) = 1. Drinfeld has proved (see [14])

Theorem 1.1. — Let E be a finite extension of ℚ_{l} and ρ : π_{1}(C) → GL_{2}(E) be a geometracally irreducible continuous l-adic representation.

Then there exists a (unique up to multiplicative constant) nonzero unramified cusp form f such that, for any closed point ν of C, T_{ν}(f) = Trρ(σ_{ν})f and U_{ν}(f) = q_{ν}^{−1} det ρ(σ_{ν})f.

This statement admits a geometric counterpart ([25]), first conjecturally extends to GL_{n} by G. Laumon ([25] and [26]) and proved by E. Frenkel-D. Gaitsgory-D. Kazhdan-K. Vilonen

Theorem 1.2. — ([25] for n = 2 and [15]) Each rank n ≥ 2 and geometrically irreducible l-adic representation of π^{1}(C) corresponds to a (bounded) complex of ℚ_{l}-sheaves K_{E} (a perverse sheaf) on the stack Bun_{C,n} of rank n vector bundles on C, which is in a certain sense cuspidal and an eigenvector for Hecke operators.

One recovers the Drinfeld theorem and its extensions by using the Grothendieck dictionary “fonctions-faisceaux” and the adelic interpretation of the vector bundles on C (given by A. Weil), the automorphic form f (unramified, cuspidal, eigenvector for the Hecke operators) being given by the following formula: for all g ∈ GL_{n}(𝔸_{F})

f(g) = χ_{K_{E}} (g) = ∑_{i≥0}(−1)^{i}Tr(σ_{g}, H_{g}^{i}(K_{E})),

where σ_{g} is the Frobenius morphism acting on the stalks at g of the cohomology sheaves H^{i}(K_{E}).

When one considers the theorem 1.2, it is natural to note that the field of definition of the curve C does not matter (this is very well explained in [18]). Then one can settle a problem in zero characteristic, closed to 1.2, which is what we called the geometric correspondence ^{(2)}. Now let C be a smooth projective curve over the field of complex numbers ℂ and let G be a reductive goup also defined over ℂ, one denotes by ^{L}G the Langlands dual of G (when G = GL_{n} one has ^{L}G = GL_{n}(ℂ)) and by Bun_{C,G} the stack of G-bundles on C.

Conjecture 1.3. — Geometric correspondence. There a natural correspondence between ^{L}G-local ℚ_{l}-systems on C and perverse Hecke eigen-ℚ_{l}-sheaves on Bun_{C,G}.

(2)Sometimes named geometric Langlands correspondence, Drinfeld-Langlands correspondence, or Beilinson correspondence; indeed it is clear that it comes from main works by Langlands, Drinfeld, Laumon and Beilinson.

(This conjecture admits a precise statement, which needs a too long preparation to be settled here.) For ^{L}G-local systems which are “opers”, this is a theorem of Beilinson and Drinfeld ([6], (5.2.6)). It has been recently solved when G = GL_{n} by E. Frenkel-D. Gaitsgory-K. Vilonen ([16]) and D. Gaitsgory ([17], see also the review [27]). Problems related to 1.3 are sometimes studied with D-modules in place of perverse sheaves, this is the same up to the Riemann-Hilbert correspondence.

We are now interested by the conjecture 1.3 when the curve C is reduced to one point, this gives the so called geometric Satake correspondence, which is a theorem.

Let G be a connected complex reductive group. The geometric Satake correspondence says that there exists a natural equivalence between the category of finite dimensional representations of the Langlands dual ^{L}G of G and a certain tensor category of perverse sheaves (or D-modules) on the (infinite) affine grassmannian of G. Proofs, or outline of proofs, using the perverse sheaves point of view, have been given by V. Ginzburg ([18]), by I. Mirković and K. Vilonen ([32]); in [6], §5.3, A. Beilinson and V. Drinfeld described the proof using D-modules. We will give in § 2.3 the precise statement of this last formulation of the Satake correspondence.

The aim of our work it to give an analog of the Satake correspondence, when D-modules are replaced by D_{σ}-modules, that means when the usual derivation is replaced by σ-derivations, for automorphisms σ of the affine grassmannian. More precisely, let K be a “big" field admitting k as subfield, let C be a smooth, projective and geometrically irreducible curve over the field K, admitting a smooth model over k, let P be a k-rational point of C and σ be an automorphism of C, defined over k, which leaves fixed the point P. The affine grassmannian \underline{Gr} of G can be interpreted as the stack of isomorphy classes of G-bundles on the infinitesimal disk around P ([3] and [21], see also our review § 2), then σ acts naturally on \underline{Gr}, which permits to define σ-derivations and D^{σ}-modules (i.e. Dmodules for σ-derivations) on \underline{Gr}. We give a common formalism for an analog of the Satake correspondence, for certain automorphisms σ, mainly

- when k = 𝔽 is a finite field, K = 𝔽^{alg} is an algebraic closure of 𝔽 and σ is the geometric Frobenius morphism of C coming from the Frobenius morphism of K with fixed field k,

- or when k = K = ℂ and σ is induced on the curve by the q-difference morphism on its field of rational functions.

This work has began following discussions with Jean-Pierre Ramis, we have had many stimulated conversations with Vadim Schechtman during its preparation. We would like to thank both here.

1.1. Notations, conventions.

All rings, algebras… will be commutative with units and their morphisms will map unit to unit (the morphism inclusion also). Let k be any field, K = k^{alg} be an algebraic closure of k and 𝕆 = K[[t]] be the ring of formal power series with coefficients in K, 𝕂 = Quot(𝕆) = K((t)).

Let G be a connected reductive group defined over K. For any commutative (with unit) K-algebra R, we set G(R[[t]]) and G(R((t))) to be respectively the R[[t]]-valued and R((t))-valued points of G.

Let C be a smooth connex projective curve defined over K, we suppose that C has a model over k, i.e. that there exists a curve C_{k} defined and smooth over k such that C_{k} ×_{Speck} SpecK = C.

For all algebra R as before, we set

C_{R} = C ×_{SpecK} SpecR and C_{R}^{∗} = (C − {P }) ×_{SpecK} SpecR;

we set also

D_{R} = SpecR[[t]] and D_{R}^{∗} = SpecR((t)),

(we delete the subscript when R = K). We have the cartesian diagram

\begin{matrix}D_{R}^{∗} & ↪ & D_{R} \\ ↓ & & ↓ \\ C_{R}^{∗} & ↪ & C_{R} \end{matrix} (2)

the image of the closed point of D_{R} being P and the horizontal arrows being inclusions.

2. The affine grassmannian

Our main referencs are [3] and [21], but some facts or informations need [35] and [30].

2.1. The functors \underline{G(𝕆)} and \underline{G(𝕂)}.— Let (K −alg) be the category of K-algebras, associative, commutative, with unit and (aff − K − Sch) be the opposite category of affine K-scheme. We denote also by K − Sch the category of K-schemes. Let E be the category of sets or groups, the two functors

\begin{aligned}\underline{G(𝕆)} : & (K − alg) & → & E \\ & R & ↦ & G(R[[t]]) \end{aligned} (3)

\begin{aligned}\underline{G(𝕂)} : & (K − alg) & → & E \\ & R & ↦ & G(R((t))) \end{aligned} (4)

are K-spaces (see [28]), this means that they are sheaves for the faithfully flat topology.

2.1.1. The K-space \underline{G(𝕆)} is a representred by an affine group scheme, indeed

\underline{G(𝕆)} = Hom_{(K−Sch)}( · , G(𝕆))

and G(𝕆) is an affine group scheme, for instance, if G = GL_{n},

GL_{n}(𝕆) = GL_{n}(K) × ∏_{i∈ℕ^{∗}} M_{n}(K),

where the products of the different schemes are over SpecK.

2.1.2. The K-space \underline{G(𝕂)} is represented by an ind-affine group, more precisely, there exists an increasing sequence (G(𝕂)_{N})_{N∈ℕ} of affine scheme such that the inclusions G(𝕂)_{N} ⊆ G(𝕂)_{N+1} are closed immersions, and, if \underline{G(𝕂)}_{N} denote the functor

\underline{G(𝕂)}_{N} : (K − alg) →  E R  ↦ Hom_{(K−Sch)}(SpecR, G(𝕂)_{N}) ,

such that

\underline{G(𝕂)} = lim_{→}(\underline{G(𝕂)}_{N})_{N∈ℕ} = lim_{→}(Hom_{(K−Sch)}( · , G(𝕂)_{N}))_{N∈ℕ},

then such that for all R

\underline{G(𝕂)}(R) = lim_{→}(\underline{G(𝕂)}_{N}(R))_{N∈ℕ} = lim_{→}(G(R((t)))_{N}) = G(R((t))).

For instance, if G = GL_{n}, for any non negative integer N let

M_{n}(𝕂)_{N} = ∏_{i≥−N} t^{i}M_{n}(K) ⊆ M_{n}(K((t))),

then

GL_{n}(𝕂)_{N} = {P, Q ∈ M_{n}(𝕂)_{N} / P Q = Id}.

In the general case, we can suppose that G is a closed subscheme of GL_{n}, for some n, and we have

G(𝕂)_{N} = G(𝕂) ∩ GL_{n}(𝕂)_{N}.

2.1.3. For any algebra R, let F_{1}(R) be the isomorphy classes of triples (E, φ, ψ), where E is a G-bundle on C_{R}, φ : C_{R}^{∗} ×_{SpecK} G ≃ E_{|C_{R}^{∗}} is a trivialisation of E over C_{R}^{∗} and ψ : D_{R}×_{SpecK} G ≃ E_{|D_{R}} is a trivialisation of the pull back E_{|D_{R}} = E ×_{C_{R}} D_{R} of E on D_{R}.

Proposition 2.1. — The two functors F_{1} and \underline{G(𝕂)}, from (R − alg) to the category of sets, are isomorphic.

Proof. — (sketch) Let (E, φ, ψ) be as above. With the diagram (2) one can define the two maps

f : D_{R}^{∗} ×_{SpecK} G → C_{R}^{∗} × G →^{φ} E ×_{C_{R}} C_{R}^{∗} → E ×_{C_{R}} D_{R}^{∗},

g : D_{R}^{∗} ×_{SpecK} G → D_{R} × G →^{ψ} E ×_{C_{R}} D_{R} → E ×_{C_{R}} D_{R}^{∗}.

One sees that f and g are two trivializations of E over D_{R}^{∗}, then g^{−1} ∘ f is an automorphism of the trivial G-bundle D_{R}^{∗} ×_{SpecK} G, which is given by an element of G(R((t))) = \underline{G(𝕂)}(R), as every R((t))-linear automorphism of the algebra of D_{R}^{∗}×_{SpecK} G. This gives a map F_{1}(R) → \underline{G(𝕂)}(R) which is obviously functorial in R.

2.2. The functor \underline{Gr} (i.e. the affine grassmannian of G).— Let T be the fpqc site of K-schemes, i.e. T is a category which objects are Kschemes and which arrows are faithfully flat, quasi compact morphisms of K-schemes. The functors \underline{G(𝕆)} and \underline{G(𝕂)} extend naturally to T. Let S be the category of scheaves on T.

Definition 2.2. — The affine grassmannian \underline{Gr} of G is, in the abelian category S, the cokernel of the inclusion morphism \underline{G(𝕆)} ↪ \underline{G(𝕂)}.

Proposition 2.3. — The affine grassmannian \underline{Gr} of G is representable by an ind-projective K-scheme: there exists an increasing sequence (Gr_{N})_{N∈ℕ} of projective K-schemes, such that the inclusion morphisms Gr_{N} ↪ Gr_{N+1} are closed immersions and such that, if Gr := ∪_{N∈ℕ}Gr_{N}, then for any K-scheme U, \underline{Gr}(U) = Hom_{K−Sch}(U, Gr).

Moreover the ind-scheme Gr is equipped with a left action of G(𝕆) such that each Gr_{N} is G(𝕆)-stable and consists of finitely many orbits of the form G(𝕆)χ, where χ is (the canonical image in Gr of) a one parameter subgroup χ : 𝔾_{m} → G. This decomposition of Gr in G(𝕆)-orbits is a stratification.

The proof is given, at least partially, in many papers (see [3], [4], [18]), it is complete in [6], § 4.5 and 7.11.2.

Remark 2.4. — The ind-scheme Gr is, in the fpqc site of ind-K-schemes, the fpqc quotient

Gr = G(𝕂)/G(𝕆)

and with the notations of 2.1.2 one has for all N

Gr_{N} = G(𝕂)_{N}/G(𝕆).

2.2.1. The group G(𝕂) acts by the left on Gr, for any γ ∈ G(𝕂) let (γ(N))_{N∈ℕ} be an increasing sequence of integers such that for any N

γ(Gr_{N}) ⊂ Gr_{γ(N)},

and we suppose moreover that γ(N) ≥ N and that γ(N) is chosen as small as possible. If γ ∈ G(𝕆), we have γ(N) = N.

2.2.2. By restriction on objects of T which are affine K-schemes, one can consider \underline{Gr} as a functor on (aff-K-Sch) (or (K-alg)).

For any algebra R, let T_{2}(R) be the isomorphy classes of pairs (E, φ), where E is a G-bundle on C_{R} and φ : C_{R}^{∗} ×_{SpecK} G ≃ E_{|C_{R}^{∗}} is a trivialisation of E over C_{R}^{∗}.

Proposition 2.5. — The two functors T_{2} and \underline{Gr}, from (K − alg) to the category of sets, are isomorphic.

The proof uses same arguments as in proposition 2.1.

2.3. The geometric Satake correspondence.— In this section, we have K = ℂ. This correspondence is given in [18], [32] and [6] § 5.3, we follow the last reference, where it is given in terms of D-modules.

The geometric Satake correspondance is between on one hand the category Rep^{L}G of finite representations the langlands dual ^{L}G and on the other hand the category of coherent holonomic D-modules on \underline{Gr}. This functor \underline{Gr} in indeed a stack (and the notion of D-modules on a stack has been clarified in [6]), it is represented by the ind-projective K-scheme Gr.

Let χ be a one parameter subgroup of G, any smooth D-module on the orbit G(𝕆)χ (see proposition 2.3) is constant ([6], lemma 5.3.2). Let P be the category of holonomic D-modules on Gr smooth on each G(𝕆)- orbit, it is semisimple (op. cit. proposition 5.3.3) and carries a canonical tensor structure (op. cit. 5.3.5, [32]). Consider G(𝕂) × Gr equipped with the following action of G(𝕆): for any (g, x) ∈ G(𝕂) × Gr and any γ ∈ G(𝕆), we set γ · (g, x) = (gγ^{−1}, γx). Denote by G(𝕂) ×_{G(𝕆)} Gr the quotient of G(𝕂) × Gr by this action. Consider the following diagram

Gr × Gr ←^{u} G(𝕂) × Gr →^{v} G(𝕂) ×_{G(𝕆)} Gr →^{m} Gr (5)

where u is the product of the canonical map G(𝕂) → Gr and of the identity map on Gr, where v is the quotient morphism and where m is the multiplication map. Let P_{1} and P_{2} be two D-modules on Gr, as u and v are locally trivial for the Zariski topology (op. cit. theorem 4.5.1), there exists a D-module P on G(𝕂) ×_{G(𝕆)} Gr such that v^{∗}P = P_{1} ⊠ P_{2}. We set

P_{1} ⊛ P_{2} = m_{∗}P (6)

and a very important result says that P_{1} ⊛ P_{2} is a D-module on Gr, that (6) defines a tensor product on P ([18], [32] and [6] theorem 5.3.6).

For P in P set h^{·}(P) = H_{DR}^{·}(P) (to define the De Rham cohomology we use the fact that Gr is an ind-projective scheme). Let \underline{Vect}_{ℂ} be the category of gradued finite dimensional ℂ-vector spaces, then we have a functor

h^{·} : P ⟶ \underline{Vect}_{ℂ}, (7)

indeed h^{·} is a tensor functor and P is a tannakian category, the isomorphism group of h^{·} is isomorphic with the Langlands dual ^{L}G of G. Then it follows from the general tannakian formalism ([10],[9] and [8]) an equivalence of categories

P ≃ Rep^{L}G,

where Rep^{L}G is viewed as a tensor category (of finite representations of LG). All these facts can be found in [18], or [32], or [6] § 5.3.

3. Topology and sheaves

The notations are now those of § 1.1.

3.1. Actions on the affine grassmannian.— 3.1.1. Two automorphisms on C.— First we suppose that the curve C is defined over ℂ, i.e. that k = K = ℂ. Let q be a complex number, q ≠ 0. Suppose that the field K(C) of rational functions over C is equipped with a q-difference automorphism σ^{♯}, it means that there exists a field extension

K(X) ↪ K(C),

which is finite (where X is an indeterminate), and an automorphism σ^{♯} of K(C) which retricted to K(X) gives the q-difference automorphism sending f(X) ∈ K(X) to f(qX). Then σ^{♯} induces an automorphism of the curve C, denoted by σ. We suppose that this automorphism leaves fixed a closed point P of C.

Now k is a finite field of characteristic p, with p^{r} elements, and K is an algebraic closure of k. Let τ be the absolute Frobenius morphism of C_{k} and denote by σ the endomorphism τ^{r} ×Id_{K} on C_{k} ×_{Speck} SpecK = C. This morphism σ is a Frobenius morphism of C, which leaves fixed all k-rational points of C. Let P be a k-rational point of C.

3.1.2. Actions on \underline{Gr}.— Let P be a closed point of C fixed under σ and denote by t_{P} a local parameter at P. We have the isomorphisms

K((t_{P})) ≃ 𝕂 and K[[t_{P}]] ≃ 𝕆

given by t_{P} ↦ t, this implies an action of σ on D and D^{∗}; we denote again by σ this last action. By scalar extension, σ acts also on C_{R}, D_{R} and D_{R}^{∗}, for any K-algebra R (on which σ acts trivially).

All these σ-action permit to construct an action on \underline{Gr}.

Lemma 3.1. — Let R be a K-algebra and Y an R-scheme equipped by a R-morphism Y → C_{R} and by an automorphism Σ compatible with σ, i.e. such that the following diagram is commutative

\begin{matrix}Y & →^{Σ} & Y \\ ↓ & & ↓ \\ C_{R} & →^{σ} & C_{R} \end{matrix}

Let E be a G-bunble on C_{R} and set

E_{σ} = E ×_{C,σ} C,

where the morphism C → C used in the product is σ. Suppose that E is trivial on Y, i.e. that E ×_{C_{R}} Y ≃ G ×_{SpecK} Y, then E_{σ} is also trivial on Y; more precisely, we have a commutative diagram

E_{σ} ×_{C_{R}} Y ≃ G ×_{SpecK} Y
ρ ↓  ↓
E ×_{C_{R}} Y ≃ G ×_{SpecK} Y

where the left vertical arrow comes from the projection E_{σ} → E and the right is Id_{G} × Σ^{−1}.

Proof. — It is easy, it suffices to remark that

E_{σ} ×_{C_{R}} Y ≃ (E ×_{C_{R}} Y) ×_{Y,Σ} Y.

It follows from this lemma that σ induces an automorphism of \underline{Gr}, denoted always σ. We do not give more details because the proof is straightforward (see § 2.2.2).

As in § 2.2.2, consider (E, φ), where E is a G-bundle on C_{R} and φ : C_{R}^{∗} ×_{SpecK} G ≃ E_{|C_{R}^{∗}} is a trivialisation of E over C_{R}^{∗}; the pair (E, φ) represents an element of \underline{Gr}(R) (see prop. 2.5), it follows from the lemma 3.1 (with C_{R}^{∗} in place of Y) that its image by σ is the isomorphy class of

(E_{σ}, ((Id_{G} × σ) ∘ φ ∘ ρ)^{−1})

(note that, as before, we denote Σ on C_{R}^{∗} by σ).

3.1.2.1. The q-difference case.— Let

\begin{matrix}𝕂 & → & 𝕂 \\ ∑_{−∞≪i} a_{i}t^{i} & ↦ & ∑_{−∞≪i} a_{i}(qt)^{i} \end{matrix}

This map induces endomorphisms (all denoted by σ) on G(𝕂), on Gr and the expected σ on \underline{Gr}.

Note that any orbit G(𝕆)χ (see proposition 2.3) is maped on itself by σ.

3.1.2.2. The Frobenius case.— Let

\begin{matrix}𝕂 & → & 𝕂 \\ ∑_{−∞≪i} a_{i}t^{i} & ↦ & ∑_{−∞≪i} a_{i}t^{ip^{r}} \end{matrix}

(recall that the fixed point P is rational over k). This map induces endomorphisms (all denoted by σ) on G(𝕂), on Gr and the expected σ on \underline{Gr}.

The image by σ of a G(𝕆)-orbit is a G(𝕆)-orbit, but in general it is not the same. We have for any integer N ≥ 0 (see proposition 2.3 and § 2.1.2) σ(Gr_{N}) ⊂ Gr_{p^{r}N}; this is a closed embedding of degree p^{r}, as the inclusion Gr_{N} ⊂ Gr_{p^{r}N}.

3.1.3. Notations.— Let (σ(N))_{N∈ℕ} be an increasing sequence of integers such that for any N

σ(Gr_{N}) ⊂ Gr_{σ(N)},

and we suppose moreover that σ(N) ≥ N and that σ(N) is chosen as small as possible. In the rest of this paper we suppose that such a sequence exists, as for the two examples above.

3.2. The topology T.— We want to define D-modules for σderivations in place of the usual one. As in the Satake correspondance, we will consider these D-modules restricted to the G(𝕆)-orbits, but in our situation, at least in the Frobenius morphism case, we cannot consider a G(𝕆)-orbit alone, indeed they have to be considerd with all their images by the σ^{n}, n ∈ ℕ; moreover the main property of these orbits is that any morphism

G(𝕆) → Gr,

which maps G(𝕆) onto an orbit, is fpqc (faithfully flat and quasi compact). These two remarks maybe explain why we define the following Grothendieck topology.

Let T be the following category:

The objects of T are the

U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U} = (σ_{U,N})_{N∈ℕ}),

where

- for any N ∈ ℕ, U_{N} is a smooth K-scheme of finite type and φ_{N} : U_{N} → U_{N+1} is a closed immersion,

- for any N ∈ ℕ, f_{N} : U_{N} → Gr_{N} is a fpqc morphism, compatible with the φ_{N}, i.e. such that

U_{N+1} →^{f_{N+1}} Gr_{N+1}
φ_{N} ↑  ∪
U_{N} →^{f_{N}} Gr_{N}

is a commutative diagram,

- for any N ∈ ℕ, σ_{U,N} : U_{N} → U_{σ(N)} is a morphism of K-schemes such that

U_{N} →^{σ_{U,N}} U_{σ(N)}  U_{N} →^{σ_{U,N}} U_{σ(N)}
f_{N} ↓ ↓ f_{σ(N)} φ_{N} ↓ ↓ φ_{N}^{σ}
Gr_{N} →^{σ} Gr_{σ(N)} U_{N+1} →^{σ_{U,N+1}} U_{σ(N+1)}

where φ_{N}^{σ} = φ_{σ(N+1)−1} ∘ ⋯ ∘ φ_{σ(N)}, are commutative diagrams (see 3.1.3),

- the ind-group G(𝕂) acts on U and this action is compatible with that of σ, this meams that for each N and each γ ∈ G(𝕂) we have the commutative diagrams (see 2.2.1)

U_{N} →^{γ·} U_{γ(N)}  U_{N} →^{γ·} U_{γ(N)}
f_{N} ↓ ↓ f_{γ(N)} φ_{N} ↓  ↓ φ_{N}^{γ}
Gr_{N} →^{γ·} Gr_{γ(N)} U_{N+1} →^{γ·} U_{γ(N+1)}

where φ_{N}^{γ} = φ_{γ(N+1)−1} ∘ ⋯ ∘ φ_{γ(N)} and where γ· is the left action by γ.

A morphism in T is the datum

u = (u_{N})_{N∈ℕ} : U ⟶ V,

where U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) and V = ((V_{N}, ψ_{N}, g_{N})_{N∈ℕ}, σ_{V}) are objects of T and each u_{N} is a morphism fpqc morphism u_{N} : U_{N} → V_{N}, compatible with all previous data, this means that we have commutative diagrams

U_{N} →^{u_{N}} V_{N}  U_{N} →^{u_{N}} V_{N}
φ_{N} ↓  ↓ ψ_{N} f_{N} ↓ ↓ g_{N}
U_{N+1} →^{u_{N+1}} V_{N+1} Gr_{N} = Gr_{N}

and

U_{N}  →^{u_{N}} V_{N}  U_{N} →^{u_{N}} V_{N}
σ_{U,N} ↓  ↓ σ_{V,N} γ· ↓  ↓ γ·
U_{σ(N)} →^{u_{σ(N)}} V_{σ(N)} U_{γ(N)} →^{u_{γ(N)}} V_{γ(N)}

where in the last diagram γ ∈ G(𝕂). We will omit often the subscript N in u_{N}, then for instance we will write now u : U_{N} → V_{N} in the preceeding diagrams.

Note that Gr is an object of T (Gr = ((Gr_{N}, ⊂, =)_{N∈ℕ}, σ)).

Let U be an object of T, a covering of U is the datum of a family (U^{i}, u^{i})_{i∈I} such that each U^{i} is an object of T and each u^{i} : U^{i} → U a morphism in T, we suppose moreover that for each N, {u^{i}(U_{N}^{i})}_{i∈I} is a (set) covering of U_{N}.

Proposition 3.2. — The category T equipped wih the preceeding notion of coverings is a Grothendieck topology.

Axioms of such a topology (and accessible perequisites) can be found for instance in [37]. The proof of the proposition is immediate, we just mention how are fiber products in T.

Lemma 3.3. — Let

U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) and V = ((V_{N}, ψ_{N}, g_{N})_{N∈ℕ}, σ_{V})

be two objects of T, then their fiber product exists in T, it is

U ×_{T} V = (((U_{N} ×_{Gr_{N}} V_{N}, φ_{N} × ψ_{N}, f_{N} × g_{N})_{N∈ℕ}, σ_{U×_{T} V} ), pr_{1}, pr_{2}),

where U_{N} ×_{Gr_{N}} V_{N} is the fiber product in the category of schemes over Gr_{N}, with projections pr_{1,N} and pr_{2,N}, where pr_{1} = (pr_{1,N})_{N∈ℕ} and pr_{2} = (pr_{2,N})_{N∈ℕ}, and where σ_{U×_{T} V} = (σ_{U,N} × σ_{V,N})_{N∈ℕ}.

3.3. Sheaves on the topology T.—

3.3.1. The structure sheaf of T.— Let O be the following abelian sheaf on T: for any object U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) of T we set

O(U) = ((O_{N}(U), φ_{N}^{♯})_{N∈ℕ}, σ_{U}^{♯}),

where O_{N}(U) = O_{U_{N}} (U_{N}) is the ring of global sections of the structural sheaf of the scheme U_{N} and where φ_{N}^{♯} and σ_{U}^{♯} = (σ_{U,N}^{♯}) are the comorphisms of φ_{N} and σ_{U} = (σ_{U,N}). Note that the sheaf O can be viewed as a sheaf with values in the category of inverse systems of K-algebras, note also that it is equipped by a G(𝕂)-action, coming from the same action on the objects U of T. The sheaf O is called the structure sheaf on T, this is an O^{p}-module, in the sense of [6], § 7.11.3 and 4 (see 3.3.4).

3.3.2. The sheaf Ô.— Let U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) be an object of T, we set

Ô(U) =^{def} lim_{⟵}_{N∈ℕ}(O_{N}(U), φ_{N}^{♯}). (8)

On the left hand side we take the pro-discrete topology, i.e. the topology induces by the discrete topology on each O_{N}(U), it is equiped with a σ-action (denoted again σ), which is induces by the one of O (see 3.3.1). Then (8) defines a sheaf on T, with values in topological K-algebras equipped by a continuous σ^{♯}-action and by a continuous G(𝕂)- action, these two actions coming from the hypotheses satisfied by the topology T.

An element a of Ô(U) will be usually written a = (a_{N}) with a_{N} in O_{N}(U).

3.3.3. O-modules.— An O-module on T is a sheaf S on T such that for any object U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) of T,

S(U) = (S_{N}(U_{N}))_{N∈ℕ},

where for each N S_{N}(U_{N}) is an O_{N}(U)-module, these structures of modules being compatible with the morphisms in T (indeed each S_{N} is a sheaf of O-modules on the fpqc site of K-schemes).

We will write often S_{N}(U) in place of S_{N}(U_{N}).

The sheaf O_{N} is itself an O-module and also for each N the sheaf O_{N} (for any object U of T, the N′-component of O_{N}(U) is {0} if N′ ≠ N and the N-th one is O_{U_{N}} (U_{N})).

3.3.4. O^{p}-modules.— An O^{p}-module on T is a sheaf which associates to each object U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) of T a sequence

P (U) = (P_{N}(U), P (φ_{N}))_{N∈ℕ},

such that (P_{N}(U))_{N∈ℕ} is an O-module and where P (φ_{N}) : P_{N+1}(U) → P_{N}(U) is an O_{N+1}(U)-morphism (P_{N}(U) is an O_{N+1}(U)-module via φ_{N}^{♯}). We suppose moreover that we have the isomorphisms of O_{N+1}(U)- modules

O_{N}(U) ⊗_{O_{N+1}(U)} P_{N+1}(U) ⟶^{Id⊗P (φ_{N})} P_{N}(U). (9)

We suppose also that the ind-group G(𝕂) acts on P, more precisely that for each integer N ≥ 0 we have a O(U_{γ(N)})-morphism (see 2.2.1)

P (γ(N)) : P (U_{γ(N)}) ⟶ P_{N}(U), (10)

P_{N}(U) being a O(U_{γ(N)})-module via γ^{♯} : O(U_{γ(N)}) → O_{N}(U).

Let U be an object of T and let T_{|U} be the category whose objects are the pairs (V, v), V being an object of T and v : V → U a morphism in T, whose morphisms w : (V, v) → (V′, v′) are morphism in T such that v′ ∘ w = v, whose coverings (V^{i}, v^{i})_{I∈I} are coverings in T with morphisms v^{i} in T_{|U}. If S is a sheaf on T, we denote by S_{|U} the sheaf that it induces on T_{|U} (i.e. its restriction to T_{|U}). An O^{p}-module P is said quasicoherent if it is locally presented, i.e. if for each object U of T their exists a covering (U^{i}, u^{i})_{i∈I} such that, for all i, their exists an exact sequence of sheaves on T_{|U_{i}}

L_{i} → L′_{i} → P_{|U_{i}} → 0,

which is a sequence of O_{|U_{i}}-modules, L_{i} and L′_{i} being free O_{|U_{i}}-modules. We denote by M^{p} the category of quasicoherent O^{p}-modules.

3.3.5. O^{!}-modules.— We continue to define O-modules as in [6], § 7.11.3 and 4. An O^{!}-module is a sheaf which associates to each object U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) of T a sequence

M(U) = (M_{N}(U), M(φ_{N}))_{N∈ℕ},

where (M_{N}(U))_{N∈ℕ} is an ′-module and where M(φ_{N}) : M_{N}(U) → M_{N+1}(U) is an O_{N+1}(U)-morphism (M_{N}(U) is an O_{N+1}(U)-module via φ_{N}^{♯}) such that we have for all N the isomorphism of O_{N+1}(U)-modules

θ_{N}(U) : M_{N}(U) ≃ Hom_{O_{N+1}(U)}(O_{N}(U), M_{N+1}(U)), (11)

which maps m ∈ M_{N}(U) to u : O_{N}(U) → M_{N+1}(U) defined by u(1) = M(φ_{N})(m) (recall that φ_{N}^{♯} is surjective).

It follows from these hypothesises

Lemma 3.4. — For all integer N, the map M(φ_{N}) is injective and its image is

ImM(φ_{N}) = {m ∈ M_{N+1}(U) / am = 0 ∀a ∈ ker φ_{N}^{♯}}.

Denote by M be the category of O^{!}-modules, it is abelian.

3.3.6. Tensor product.— Let M and P be objects of M and M^{p} respectively. let U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) be an object of T. We have for any integer N ≥ 0

Hom_{O_{N+1}(U)}(O_{N}(U), M_{N+1}(U) ⊗_{O_{N+1}(U)} P_{N+1}(U))
≃ Hom_{O(U_{N+1})}(O_{N}(U), M_{N+1}(U)) ⊗_{O_{N+1}(U)} P_{N+1}(U)
≃  M_{N}(U) ⊗_{O_{N+1}(U)} P_{N+1}(U)
≃  M_{N}(U) ⊗_{O_{N} (U)} P_{N}(U)

the first isomorphism uses the fact that φ_{N}^{♯} is surjective, the two others comes from (3.3.5) and (3.3.4). It follows that the rule

U ↦ (M ⊗_{O} P )(U) =^{def} (M_{N}(U) ⊗_{O_{N} (U)} P_{N}(U), ρ_{N})_{N∈ℕ},

where ρ_{N} is the composition of the two morphisms

\begin{matrix}M_{N}(U) ⊗_{O_{N} (U)} P_{N}(U) & ⟶^{∼} & M_{N}(U) ⊗_{O_{N+1}(U)} P_{N+1}(U) \\ M_{N}(U) ⊗_{O_{N+1}(U)} P_{N+1}(U) & ⟶^{M(φ_{N})⊗Id} & M_{N+1}(U) ⊗_{O_{N+1}(U)} P_{N+1}(U) \end{matrix}

defines a tensor product

M × M^{p} ⟶ M.

4. Derivations and differentials.

4.1. σ-derivations.—

4.1.1. σ-derivations and O^{!}-modules.— let M be an O^{!}-module and U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) be an object of T. Let N ≥ 0 be an integer. A σ-derivation

d_{N}(U) : Ô(U) ⟶ M_{N}(U)

defined on Ô(U) and with values in M_{N}(U) is a k-linear continuous morphism, M_{N}(U) being equipped with the discrete topology (the field k has been introduced in 1.1 and 3.1), satisfying the following rule: for all a and b in Ô(U),

d_{N}(U)(ab) = σ(a)_{N}d_{N}(U)(b) + b_{N}d_{N}(U)(a)

(for the notations see 3.3.2).

Let \underline{Der}_{σ,N}(M)(U) be the set of these derivations, this is a O(U)- module, via the canonical morphism O(U) → O_{N}(U) and the O_{N}(U)- structure on M_{N}(U).

Let M be a O^{!}-module. For any object U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) of T let

\underline{Der}_{σ}(M)(U) = (\underline{Der}_{σ,N}(M)(U), ζ_{N})_{N∈ℕ}, (12)

where

ζ_{N} : \underline{Der}_{σ,N}(M)(U) → \underline{Der}_{σ}(M)(U)_{N+1}

maps δ to M(φ_{N}) ∘ δ. This defines a sheaf on T, called the derivation sheaf with values in M, its sections are the σ-derivations.

Proposition 4.1. — Let M be a O^{!}-module, then \underline{Der}_{σ}(M) is also a O^{!}-module.

Proof. — Let U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) be an object of T. During this proof we set O_{N} = O_{N}(U), M_{N} = M_{N}(U), D_{N} = \underline{Der}_{σ,N}(M)(U), etc. We have to prove that for all N we have the isomorphism

D_{N} ≃ Hom_{O_{N+1}}(O_{N}, D_{N+1}),

given by the map which sends δ ∈ D_{N} to the map u such that u(1) = M(φ_{N}) ∘ δ (see 12). The lemma 3.4 shows that this map is well defined. We construct the inverse map: let u ∈ Hom_{O_{N+1}}(O_{N}, D_{N+1}), then au(1) = 0 for all a ∈ ker φ_{N}^{♯}, it follows that u(1) takes values in ImM(φ_{N}) (op. cit.), as M(φ_{N}) is injective, this means that their exists δ ∈ D_{N} such that u(1) = M(φ_{N}) ∘ δ.

Let M be an O^{!}-module, for any object U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) of T, we set

Der_{σ}(M)(U) = lim_{⟶}_{N∈ℕ}(\underline{Der}_{σ,N}(M)(U), ζ_{N}),

this defines the sheaf Der_{σ}(M) on T, which in an obvious sense an Ômodule. Sections of this sheaf are called global σ-derivations with values in M.

4.1.2. σ-derivations and O^{p}-modules.— The definition of a σ-derivation with values in an O^{p}-module P and its main properties are the same as before. Let U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) be an object of T, we have for any N a map

η_{N} : \underline{Der}_{σ,N+1}(P) → \underline{Der}_{σ,N}(M),

which is the composition on the left by P_{N}(φ_{N}^{♯}), then

Proposition 4.2. — Let M be a O^{p}-module, then \underline{Der}_{σ}(P) is also a O^{p}-module.

4.2. σ-differentials.—

4.2.1. The sheaves I^{r}. — Consider the sheaf Ô ⊗_{K} Ô (see 3.3.2) associated to the presheaf on T

U ↦ Ô(U) ⊗_{K} Ô(U)

(U is an object of T), it is an Ô-module by the right.

Let r be a non negative integer and

Φ^{r} : Ô ⊗_{K} Ô → Ô

be the Ô-linear map associated to the morphism of presheaves defined by

\begin{aligned}Φ̃^{r}(U) : & Ô(U) ⊗_{K} Ô(U) & → & Ô(U) \\ & a ⊗ b & ↦ & σ^{r}(a)b \end{aligned}

Let I^{r} be the kernel of Φ^{r}, it is a ideal of Ô⊗_{K} Ô. Note that I^{r} is locally generated by the a ⊗ 1 − 1 ⊗ σ^{i}(a), where a is a local section of Ô.

4.2.2. The sheaf Ω_{σ}^{1}. — We denote by Ω_{σ}^{1} the quotient sheaf

Ω_{σ}^{1} = I^{0}/I^{0}I^{1}

and let

d^{0} : Ô ⟶ Ω_{σ}^{1}

be the map induces by a ↦ a ⊗ 1 − 1 ⊗ a. The structure of Ô-module by the right on Ô ⊗_{K} Ô induces such a structure on Ω_{σ}^{1}, it is a topological Ô-module, for the quotient topology. Note that Ω_{σ}^{1} is as Ô-module generated by the image of d.

Let M be an O^{!}-module and let Hom_{Ô,cont}(Ω_{σ}^{1}, M) be the sheaf on T

U ↦ Hom_{Ô,cont}(Ω_{σ}^{1}, M)(U) =^{def} (Hom_{Ô(U),cont}(Ω_{σ}^{1}(U), M_{N}(U)))_{N∈ℕ}, (13)

where elements of Hom_{Ô(U),cont}(Ω_{σ}^{1}(U), M_{N}(U)) are continuous Ô(U)- morphisms, for the discrete topology on M_{N}(U). Moreover this sheaf is equiped with morphisms

Hom_{Ô(U),cont}(Ω_{σ}^{1}(U), M_{N}(U)) ⟶ Hom_{Ô(U),cont}(Ω_{σ}^{1}(U), M_{N+1}(U))

induced by the M(φ_{N}) (see 3.3.5) and then this is an O^{!}-module.

Proposition 4.3. — Let M be an O^{!}-module, then,

\underline{Der}_{σ}(M) ≃ Hom_{Ô,cont}(Ω_{σ}^{1}, M)

functorially in M.

The proof is straightforward. It follows from the propositions 4.3 and 4.1 that for any O^{!}-module, Hom_{Ô,cont}(Ω_{σ}^{1}, M) is also a O^{!}-module.

Remark 4.4. — The sheaf Ω_{σ}^{1} is equipped with an action of σ, coming from that of O (and denoted again by σ^{♯}). Indeed, Ω_{σ}^{1} is locally generated by the set {(d^{0}a)b}, where a and b run through local sections of O, and we have σ^{♯}(d^{0}a) = d^{0}σ^{♯}(a).

Remark 4.5. — Let a be a local section of the sheaf Ô, we set d′(a) = a ⊗ 1 − 1 ⊗ a, this is a local section of the sheaf I^{0}. Let a and b be local section of Ô, we have in I^{0}

(a ⊗ 1)d′(b) = d′(ab) − d′(a)(1 ⊗ b),

it follows that Ω_{σ}^{1} has a left structure of Ô-module, which is locally given by

(a, ω) ↦ ωσ^{♯}(a),

where a and ω are local sections of respectively Ô and Ω_{σ}^{1}. Then our notions of derivations and differentials are closed for those of Y. André (see [2]).

Remark 4.6. — Let U be an object of T, M be an O^{!}-module and u ∈ Hom_{Ô,cont}(Ω_{σ}^{1}, M)(U), u = (u_{N})_{N∈ℕ} (see 13). It follows from 4.5 that we have locally, for sections a and b of Ω_{σ}^{1}(U):

u_{N}((a ⊗ 1)d(b)) = σ(a)_{N}u_{N}(b)

(see 4.3) and recall that by definition u is Ô-linear “by the right”, indeed this means that sections of Hom_{Ô,cont}(Ω_{σ}^{1}, M) are sequilinear, in the sense of [2].

4.2.3. The sheaf Θ. — Consider the sheaf

(14) Θ = Hom_{Ô,cont}(Ω_{σ}^{1}, O)

on T, defined by

U ↦ Hom_{Ô,cont}(Ω_{σ}^{1}, O)(U) =^{def} (Hom_{Ô(U),cont}(Ω_{σ}^{1}(U), O_{N}(U)))_{N∈ℕ},

where the topology on O_{N}(U) is the discrete one (recall that O_{N} is the O-module defined by O_{N}(U) = O_{U_{N}} (U_{N})) . Set, for any object U of T and any N,

Θ_{N}(U) = Hom_{Ô(U),cont}(Ω_{σ}^{1}(U), O_{N}(U)),

This sheaf Θ_{N} is naturally an O_{N}-module and we have

Θ = (Θ_{N})_{N∈ℕ}, (15)

the maps φ_{N} : U_{N} → U_{N+1} induce maps Θ_{N}(U) → Θ_{N+1}(U), then

Proposition 4.7. — The sheaf Θ is an O^{p}-module.

The G(𝕂)-action is induced by this one on O. The proof is obvious. The sheaf Θ looks like “the tangent sheaf at T ”.

Proposition 4.8. — The sheaf Θ is a torsion free O-module of profinite type, more precisely, each sheaf θ_{N} is a torsion free O_{N}-module of finite type.

Proof. — For any non negative integer N let 𝔄_{N} be the kernel of the projection Ô → O_{N}. Let I_{N}^{0} be the subsheaf of I^{0} generated locally by the a ⊗ 1 − 1 ⊗ a, where a is a local section of 𝔄_{N}, let s be the canonical morphism I^{0} → Ω_{σ}^{1} and set

Ω_{σ,N}^{1} = \frac{Ω_{σ}^{1}}{Ω_{σ}^{1}𝔄_{N} + s(I_{σ(N)}^{0})}

(σ(N) is defined in §3.1.3), this sheaf can be viewed as an O_{N}-module.

Lemma 4.9. — There is a natural isomorphism of O_{N}-modules

Θ_{N} ≃ Hom_{O_{N}} (Ω_{σ,N}^{1}, O_{N}).

Proof. — Let u ∈ Θ_{N}, we have to prove that the kernel of u contains Ω_{σ}^{1}𝔄_{N} + s(I_{σ(N)}^{0}). Let a and b be local sections of Ô, we have in Ω_{σ}^{1}

(d^{0}a)(b − σ^{♯}(b)) = (d^{0}b)(a − σ^{♯}(a)),

then if a ∈ 𝔄_{σ(N)}

(d^{0}a)(b − σ^{♯}(b)) ∈ Ω_{σ}^{1}𝔄_{N},

as U_{N} is a locally integral scheme, this means that u(d^{0}(a) = 0.

It follows from this lemma that Θ_{N} is locally finitely generated over O_{N}. Let U = (U_{N})_{N∈ℕ} be an object of T, one can suppose that U_{σ(N)} is affine, U_{σ(N)} = SpecK[t′_{1}, ⋯, t′_{r}], where t′_{1}, ⋯, t′_{r} are images of sections t_{1}, ⋯, t_{r} of Ô(U). It is clear that Ω_{σ,N}(U) is generated over O_{N}(U) by d^{0}t_{1}, ⋯, d^{0}t_{r}.

Now we prove that Θ_{N} is torsion free over O_{N}. Let U = (U_{N})_{N∈ℕ} be an object of T, up to the lemma we can consider Θ_{N} as a sheaf on U_{N} and we have to prove that any section of Θ_{N} over U_{N} does not vanish on a whole open subset of U_{N}, but this is a consequence of the fact that U_{N} is smooth: let W be an open subset of U_{N}, the restriction O_{N}(U_{N}) → O_{N}(W) is injective, then we have an injective map Θ_{N}(U_{N}) ↪ Θ_{N}(W).

Proposition 4.10. — Let M be a torsion free O^{!}-module of pro-finite type (i.e. M = (M_{N})_{N∈ℕ} and each M_{N} is torsion free and of finite type over O_{N}), then we have the isomorphism of O^{!}-modules

Hom_{Ô,cont}(Ω_{σ}^{1}, M) ≃ M ⊗_{O} Θ

(on the right hand side this is the tensor product defined in § 3.3.6).

Proof. — We have a canonical map

Ψ : M ⊗_{O} Θ ⟶ Hom_{Ô,cont}(Ω_{σ}^{1}, M),

which is defined locally by (the morphism of presheaves)

M_{N}(U) ⊗_{O_{N} (U)} Θ(U)_{N} ⟶ Hom_{Ô(U)_{N} ,cont}(Ω_{σ}^{1}(U), M_{N}(U))
m ⊗ v  ⟼  (ω ↦ v(ω)m) (16)

We prove that it is an isomorphism.

Let U = (U_{N})_{N∈ℕ} be an object of T, we can suppose that U_{N} is affine, let x be a closed point of U_{N} and denote by K(x) the residue field at x, i.e. the field K equipped with the canonical structure O_{N}(U) → K = K(x). The same arguments as in lemma 4.9 show that

Hom_{Ô(U)_{N} ,cont}(Ω_{σ}^{1}(U), M_{N}(U)) = Hom_{O_{N} (U)}(Ω_{σ,N}^{1}(U), M_{N}(U))

is of finite type over O_{N}(U) = O_{N}(U_{N}), then, for any closed point x of U_{N}, the map of 16 gives an isomorphim

M_{N}(U)⊗_{O_{N} (U)}Θ(U)_{N}⊗_{O_{N} (U)}K(x) ≃ Hom_{O_{N} (U)}(Ω_{σ,N}^{1}(U), M_{N}(U))⊗_{O_{N} (U)}K(x).

This implies the expected isomorphism, because all O_{N}(U)-modules appearing here are of finite type.

4.3. The De Rham complex. — Condider the tensor algebra ⨂Ô Ω_{σ}^{1} and let J be its ideal locally generated by

{d^{0}σ(a) ⊗ d^{0}b + d^{0}b ⊗ d^{0}a},

where a and b are local sections of Ω_{σ}^{1}. Let

Ω_{σ}^{·} =^{def} ( ⨂ ; Ô Ω_{σ}^{1})/J ,

it is a gradued algebra, we denote by Ω_{σ}^{r} the group of its homogeneous elements of degree r.

Proposition 4.11. — There exists a map of graded ring

d^{·} : Ω_{σ}^{·} ⟶ Ω_{σ}^{·+1}

which satisfies locally the following rule: let ω_{1} and ω_{2} be local sections of Ω_{σ}^{r} and Ω_{σ}^{s} respectively, then

d^{r+s}(ω_{1}ω_{2}) = (d^{r}ω_{1})ω_{2} + (−1)^{r}σ(ω_{1})d^{s}ω_{2}

(see remark 4.4); (Ω_{σ}^{·}, d^{·}) is a complex.

Proof. — We use the method of [7], Ch. 3, § 10, n^{∘}9.

(1) There exists a map

d^{1} : Ω_{σ}^{1} ⟶ Ω_{σ}^{2}

locally defines by

ad^{0}b ⟼ (d^{0}a)(d^{0}b),

where a and b are local sections of Ô. The proof of this fact is easy. (2) Let A be an Ô-algebra and

f : Ω_{σ}^{1} ⟶ A

be an Ô-linear map such that, locally, for local sections a and b of Ô,

f(d^{0}σ(a))f(d^{0}b) + f(d^{0}b)f(d^{0}a),

then there exists a morphism of Ô-algebras

g : Ω_{σ}^{·} ⟶ A

which extends f.

The proof is also easy, g is locally defined on Ω_{σ}^{r} by

g(ω_{1} ⋯ ω_{r}) = f(ω_{1}) ⋯ f(ω_{r})

where ω_{1}, ⋯, ω_{r} are local sections of Ω_{σ}^{1}.

Consider on A = Ω_{σ}^{·} ⊕ Ω_{σ}^{·} the multiplication defined locally by the following formula: let α, β, α′ and β′ be local sections of Ω_{σ}^{·}, α being homogeneous,

(α, β)(α′, β′) = (αα′, (−1)^{deg α}σ(α)β′ + βα′)

(σ(α) is defined in the remark 4.4), then A is a ring, with (1, 0) as unit. Let h : Ô → A be the map locally defined by a ↦ (a, d^{0}a) (a is a local section of Ô), this is a morphism of rings with units. Let f : Ω_{σ}^{1} → A be the map locally defined by f(ω) = (ω, d^{1}ω) (see part (1) of this proof, ω is a local section of Ω_{σ}^{1}). The map h makes A an Ô-algebra and a direct calculation proves that f is then a morphism of Ô-modules, which satisfies all the hypotheses of the part (2) above. Let g : Ω_{σ}^{·} → A be the map given by (2), a direct calculation again proves that pr_{1} ∘ g is the identity map on Ω_{σ}^{·}, we set g = (Id, d^{·}), then d^{·} is the expected map.

5. D_{σ}-modules

In this section we want to define a notion of D-modules for σderivations and a notion of holonomy for these modules. Holonomic D-modules for this sort of derivations has been previously defined by Loeser and Sabbah ([31]), our definitions work also in the situation they have studied.

5.1. The sheaves P_{σ}^{·}. — We use the same method as in [1]. Let r ≥ 1 be a non negative integer, set

P_{σ}^{r} = Ô ⊗_{K} Ô/(I^{0}I^{1} ⋯ I^{r})

and let

δ^{r} : Ô ⟶ P_{σ}^{r}

be the map locally defined by a ↦ a⊗1. We say that P_{σ}^{r} is the sheaf on T of σ-differential principal parts of degree r, it is a topological Ô-module, for the quotient topology (recall that, as Ô ⊗_{K} Ô, I_{σ}^{r} is an Ô-modules “by the right”).

Let r ≥ 1 and s ≥ 1 be two integers, consider the tensor product

P_{σ}^{r} ⊗Ô P_{σ}^{s},

it is an Ô-module via its right factor, but the tensor product is for the Ô-module structure of P_{σ}^{r} and the Ô-module structure through δ^{s} on P_{σ}^{s}.

Lemma 5.1. — There exists an Ô-morphism

P_{σ}^{r+s} ⟶ P_{σ}^{r} ⊗Ô P_{σ}^{s},

given locally by a ⊗ b ↦ (a ⊗ 1) ⊗ (1 ⊗ b).

Proof. — Consider the map

φ : Ô ⊗_{K} Ô ⟶ (Ô ⊗_{K} Ô) ⊗Ô (Ô ⊗_{K} Ô)

locally defined by a ⊗ b ↦ (a ⊗ 1) ⊗ (1 ⊗ b), we prove now that

φ(I^{0} ⋯ I^{r+s}) ⊆ (I^{0} ⋯ I^{r}) ⊗Ô (Ô ⊗_{K} Ô) + (Ô ⊗_{K} Ô) ⊗Ô (I^{0} ⋯ I^{s}). (17)

All following computations are local (or with sections of tensor product presheaves…), we omit to write objects of T on which sections are defined. For any integer i, 0 ≤ i ≤ r + s, set

Δ_{i} = {a ⊗ 1 − 1 ⊗ σ^{i}(a) / a ∈ Ô},

this is a set of generators of I^{i} and we have for all j, 0 ≤ j ≤ i,

φ(Δ_{i}) ⊆ Δ_{j} ⊗Ô (Ô ⊗_{K} Ô) + (Ô ⊗_{K} Ô) ⊗Ô Δ_{i−j} (18)

as it is proved by the following formulae

φ(a ⊗ 1 − 1 ⊗ σ^{i}(a)) = (a ⊗ 1) ⊗ (1 ⊗ 1) − (a ⊗ 1) ⊗ (1 ⊗ σ^{i}(a))
=  (a ⊗ 1 − 1 ⊗ σ^{j}(a)) ⊗ (1 ⊗ 1)
+ (1 ⊗ 1) ⊗ (σ^{j}(a) ⊗ 1 − 1 ⊗ σ^{i}(a)).

Let n be a non negative integer and for 0 ≤ i ≤ n let

I(n − i, i − 1) = ( ∏_{0≤j≤n−i} I^{j}) ⊗Ô ( ∏_{0≤j≤i−1} I^{j})

where ∏ I^{j} = Ô ⊗Ô Ô when j describes the empty set; let w_{i} be an element of φ(Δ_{i}), we prove

∏_{0≤i≤n} w_{i} ∈ ∑_{0≤i≤n+1} I(n − i, i − 1),

this formula implies (17), it is proved by induction: we write, following (18), w_{n+1} = u_{n+1−i} + v_{i}, with u_{n+1−i} ∈ Δ_{n+1−i} ⊗Ô (Ô ⊗_{K} Ô) and v_{i} ∈ (Ô ⊗_{K} Ô) ⊗Ô Δ_{i}, we have

∏_{1≤i≤n+1} w_{i} ∈ ∑_{0≤i≤n+1} I(n − i, i − 1)w_{n+1},

then

∏_{1≤i≤n+1} w_{i} ∈ ∑_{0≤i≤n+1} I(n − i, i − 1)(u_{n+1−i} + v_{i})

∏_{1≤i≤n+1} w_{i} ∈ ∑_{0≤i≤n+1} I(n + 1 − i, i − 1) + ∑_{0≤i≤n+1} I(n − i, i)

which gives the expected formula.

5.2. The sheaves D_{σ}^{·}. — We continue to use the same method as in [1]. As before, for any non negative integer, let O_{N} be the sheaf such that O_{N}(U) = O_{U_{N}} (u_{N}), where U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) is an object of T. For any integer r ≥ 0, let

D_{σ,N}^{r} = Hom_{Ô,cont}(P_{σ}^{r}, O_{N})

and

D_{σ}^{r} = (Hom_{Ô,cont}(P_{σ}^{r}, O_{N}))_{N∈ℕ} = (D_{σ,N}^{r})_{N∈ℕ},

where for any object U of T, O_{N}(U) is equipped with the discrete topology. The maps φ_{N}^{♯} induce morphisms

D_{σ,N+1}^{r} ⟶ D_{σ,N}^{r}.

The sheaf D_{σ}^{r}, equipped with these morphisms, is called the sheaf of σ-differential operators of degree ≤ r.

Proposition 5.2. — The sheaves D_{σ}^{r} are O^{p}-modules.

The proof is clear. Let s ≥ r ≥ 0 be integers, the canonical map P_{σ}^{s} → P_{σ}^{r} induces a canonical injective morphism D_{σ}^{r} ↪ D_{σ}^{s}, these map permit to define

D_{σ} = lim_{⟶}_{r∈ℕ}D_{σ}^{r}

which is, as a consequence of lemma 5.1, a Ô-algebra:

Proposition 5.3. — (Composition of σ-differential operators) Let r and s be non negative integers, then there is a morphism

D_{σ}^{r} ⊗Ô D_{σ}^{r} ⟶ D_{σ}^{r+s}.

In other words, D_{σ} is a Ô-algebra.

5.2.1. The sheaves D_{σ}^{·} and the O^{!}-modules;— Let r and N as before and let M be an O^{!}-module. Consider the sheaf

\underline{D}_{σ}^{r}(M) = Hom_{O,cont}(P_{σ}^{r}, M) = (Hom_{Ô,cont}(P_{σ}^{r}, M_{N}))_{N∈ℕ}

(see 13) where, for each object U of T, elements of Hom_{Ô,cont}(P_{σ}^{r}, M_{N})(U)

are continuous Ô(U)-morphisms, for the discrete topology on M_{N}(U). Moreover this sheaf is equiped with morphisms

Hom_{Ô,cont}(P_{σ}^{r}, M_{N})(U) ⟶ Hom_{Ô,cont}(P_{σ}^{r}, M_{N+1})(U)

induced by the M(φ_{N}) (see 3.3.5) and we see that Proposition 5.4. — Let M be an O^{!}-module, then the sheaf \underline{D}_{σ}^{r}(M) is also an O^{!}-module.

We have also a property closed to proposition 4.10, the proof being almost the same:

Proposition 5.5. — Let M be a torsion free O^{!}-module, then we have the isomorphism of O^{!}-modules

\underline{D}_{σ}^{r}(M) ≃ M ⊗_{O} D_{σ}^{r}

(on the right hand side this is the tensor product defined in § 3.3.6).

5.2.2. The sheaves D_{σ}^{·} and the O^{p}-modules.— In this paragraph we use a method coming from [5]. Let P be an O^{p}-module. Set \underline{D}_{σ}^{−1}(P) = \{0\} and for any integer r ≥ 0 let \underline{D}_{σ}^{r}(P) be the subsheaf of End_{k}(P) of sections f which locally satisfy the rule

f(a·) − (σ^{♯})^{r}(a)f(·) ∈ \underline{D}_{σ}^{r−1}(P)

for any section a of O; more precisely, for any object U of T, \underline{D}_{σ}^{r}(P)(U) is the set of f ∈ End_{k}(P)(U) such that, for each V → U in T and for each a ∈ O(V),

f |_{V} (a·) − (σ^{♯})_{V}^{r} (a)f |_{V} (·) ∈ \underline{D}_{σ}^{r−1}(P)(V).

Proposition 5.6. — For any non negative integer r we have an isomorphism of sheaves D_{σ}^{r} ≃ \underline{D}_{σ}^{r}(O).

Proof. — We make a confusion between the sheaf Ô and the inverse system of sheaves O. The map D_{σ}^{r} → \underline{D}_{σ}^{r}(O) is D ↦ D ∘ δ^{r} (see § 5.1). By the inverse map, the image of a section f of \underline{D}_{σ}^{r}(O) is locally defined by δ^{r}(a) ↦ f(a) (a is a section of O).

5.3. D_{σ}-modules. — A D_{σ}-module P is an O^{p}-module with a structure of module over the ring D_{σ}. This means that, if we denote by P_{N} the sheaf U ↦ P_{N}(U), P_{N} is equipped with an action of the additive group D_{σ,N}^{r}, for all N and r, which is compatible with the inverse systems D_{σ}^{r} and P, and, with the proposition 5.3, which gives a D_{σ}-module structure on P.

5.3.1. Some words about holonomy.— We do not have a complete notion of “σ-holonomy”, we explain now how maybe one can find it.

Consider the analog for σ-derivations of the Weyl algebra, namely

W_{σ} = K[z_{1}, ⋯, z_{n}, ∂_{1}, ⋯, ∂_{n}],

with for all i and j, i ≠ j,

∂_{i}z_{i} = σ(z_{i})∂_{i} + 1 and ∂_{i}z_{j} = z_{j}∂_{i},

σ being an endomorphism of the algebra K[z_{1}, ⋯, z_{n}]. For any D ∈ W_{σ} and any a ∈ K[z_{1}, ⋯, z_{n}] set [D, a]_{r} = Da−σ^{r}(a)D. With our previous definition of σ-differential operators (see § 5.2.2), we see that D ∈ D_{σ}^{r} if and only if for all a_{0}, ⋯, a_{r} ∈ K[z_{1}, ⋯, z_{n}],

[[ ⋯ [[D, a_{r}]_{r}, a_{r−1}]_{r−1}, ⋯]_{2}, a_{1}]_{1}, a_{0}]_{0} = 0,

then D_{σ}^{r} is the set of (finite sums) ∑_{α,β} a_{α,β}z^{α}∂^{β}, where α = (α_{1}, ⋯, α_{n}) and β = (β_{1}, ⋯, β_{n}) go through ℕ^{r}, and | β |= β_{1} + ⋯ + β_{n} ≤ r. Set

Gr(W_{σ}) = ⊕_{r∈ℕ}D_{σ}^{r+1}/D_{σ}^{r},

it is not a commutative algebra and we cannot define a notion of characteristic variety for a W_{σ}-module. We define holonomic W_{σ}-module whithout using such a variety.

Let P be a W_{σ}-module of finite type and let dp_{W_{σ}}(P) be the projective dimension of P, set

j(P) = max{j / Ext_{W_{σ}}^{j}(M, W_{σ}) ≠ 0},

the following lemma is wellknown

Lemma 5.7. — dp_{W_{σ}}(P) = j(P).

Definition 5.8. — Let P be a W_{σ}-module of finite type, the defect of P is d(P) := 1 − j(P)/n.

We expect the following result

Proposition 5.9. — Let P be a W_{σ}-module of finite type, then d(P) ≥ 0.

Definition 5.10. — Let P be a W_{σ}-module of finite type, it is said holonomic if d(P) = 0.

All these properties and definitions correspond to the notion of holonomy in the classical case.

5.3.2. The De Rham complexes of some D_{σ}-module.— In the classical case any smooth D-module on an orbit G(𝕆)χ (see proposition 2.3) is constant (it is proved in [6], lemma 5.3.2), this explain the following definitions.

5.3.2.1. Elementary objects.— Recall that the orbits in Gr under the left action by G(𝕆) are of the form G(𝕆)χ, where χ describes a subset Ξ of the subgroup of one parameter subgroups of G (see proposition 2.3). Let U^{ω} = ((U_{N}^{ω}, φ_{N}^{ω}, f_{N}^{ω})_{N∈ℕ}, σ_{U^{ω}}) be the following object of T: for any integer N ≥ 0 let Ξ_{N} be the set of χ ∈ Ξ such that G(𝕆)χ ⊂ Gr_{N}, for any χ ∈ Ξ_{N} let U_{N,χ}^{ω} = G(𝕆), and set

U_{N}^{ω} = ∐_{χ∈Ξ_{N}} U_{N,χ}^{ω} ,

let φ_{N}^{ω} be the inclusion U_{N}^{ω} ⊂ U_{N+1}^{ω} and f_{N}^{ω} : U_{N}^{ω} → Gr_{N} be the morphism which maps canonically U_{N,χ}^{ω} to G(𝕆)χ. Left actions by G(𝕆), G(𝕂) and the map σ_{U^{ω}} are obvious.

Let V be an object of T which admits a covering (V^{i} → V)_{i∈I} where each V^{i} = ((V_{N}^{i}, ψ_{N}^{i}, g_{N}^{i})_{N∈ℕ}, σ_{V^{i}}) is such that V_{N}^{i} is Zariski open in U_{N}^{ω}, ψ_{N}^{i}, g_{N}^{i} and σ_{V^{i}} are the restriction of φ_{N}^{ω}, f_{N}^{ω} and σ_{U^{ω}} respectively. We say that V is an elementary object of T.

Let P be an O^{p}-module on T, let V be an object of T and denote by T |_{V} the restricted topology T to V, by P |_{V} the sheaf on T |_{V} induces by P. We say that the sheaf P is O-free of pro-finite type on V if P |_{V} is O-free on T |_{V} and if each (P |_{V})_{N} is of finite type over O_{N}.

Definition 5.11. — We say that a coherent D_{σ}-module P satisfies the property (H) if it is O-free of pro-finite type on every elementary object of T.

Proposition 5.12. — Let P be a D_{σ}-module sastisfying the property (H), then for each elementary object V of T there exists a map of sheaves of graded groups on T |_{V}, of degree 1,

∇^{·} |_{V}: P ⊗Ô Ω_{σ}^{·} ⟶ P ⊗Ô Ω_{σ}^{·+1}

such that, for any integer r ≥ 0, for local sections z and ω of P and Ω_{σ}^{r} respectively,

∇^{r}(ω ⊗ z) = d^{r}(ω) ⊗ z + (−1)^{r}σ(ω)∇^{0}(z);

∇^{·} |_{V} is functorial in V, it is an integrable connection (i.e. ∇^{1} |_{V} ∘∇^{0} |_{V} = 0).

This is a consequence of the definition 5.11 and of the De Rham complex for O, see proposition 4.11.

6. A tannakian category

6.1. The symmetric product.— Let U be an elementary object of T, we consider the ind-scheme

(G(𝕂) ×_{Gr} U) ×_{SpecK} U

on which we consider the topology

T^{(U)} =^{def} (G(𝕂) ×_{Gr} (T |_{U})) ×_{SpecK} (T |_{U})

and the two morphisms of topology, with values in T |_{U}, coming from the projections (over SpecK). This topology is equipped with the action of G(𝕆) defined by the formula,

γ((g, x), y) = ((gγ^{−1}, x), γy),

where γ ∈ G(𝕆), g ∈ G(𝕂) and x, y ∈ V and V is an object of T |_{U}.

Let P and Q be two D_{σ}-modules, then on the topology T^{(U)}, we have the sheaf

(P ⊠ Q)^{G(𝕆)}

of G(𝕆)-invariant sections of the sheaf P ⊠ Q. Let

m_{U} : (G(𝕂) ×_{Gr} U) ×_{SpecK} U ⟶ U

be the map defined (with the previous notations) by m((g, x), y) = gy. Set (P ⊛ Q) |_{U}= (m_{U})_{∗}((P ⊠ Q)^{G(𝕆)}), these sheaves glue, to give a sheaf on T, denoted by P ⊛ Q.

Proposition 6.1. — Let 𝔓 be the category of D_{σ}-modules satisfying the hypothesis (H) of definition 5.11, then ⊛ is a symmetric product on 𝔓.

6.2. The fiber functor.— let P be a D_{σ}-module satisfying the hypothesis (H) of definition 5.11 and let

P → P ⊗_{O} Ω_{σ}^{1} → P ⊗_{O} Ω_{σ}^{2} ⋯

be its De Rham complex on elementary objects (see 5.12), this is a complex of inverse systems of sheaves on which we apply the functor Γ(Gr, ·) of global sections on Gr. We obtain a complex of inverse systems of O(Gr)-modules

Γ(Gr, P) → Γ(Gr, P ⊗_{O} Ω_{σ}^{1}) → Γ(Gr, P ⊗_{O} Ω_{σ}^{2}) ⋯,

take an injective resolution in the category of such complexes, thus we obtain a bicomplex of inverse systems of O(Gr)-modules, which cohomology ℍ^{·}(Gr, P) is the De Rham cohomology on Gr of the D_{σ}-module P. Note that each ℍ^{r}(Gr, P) is an inverse system of O(Gr)-modules.

Remark 6.2. — We have seen that the complex (Ω_{σ}^{·}, d^{·}) is equipped with a morphism σ^{♯} of degree 0 (for instance locally defined on Ω_{σ}^{1} by σ^{♯}(d^{0}a) = d^{0}σ^{♯}(a), where a is a local section of O), this implies the existence of a morphism of degree 0 on the De Rham complex of P, thus of a morphism, denoted again by σ^{♯}, on each inverse system ℍ^{r}(Gr, P).

Definition 6.3. — Let 𝔓_{c} be the category of D_{σ}-modules which satisfy the hypothesis (H) of proposition 5.11 and which moreover are submited to the following condition: if P is in 𝔓_{c} and if U = ((U_{N}, φ_{N}, f_{N})_{N∈ℕ}, σ_{U}) is an object of T, then the support of P on U is determined by finitely many U_{N}, more precisely, there exists an integer N_{0} = N(P, U) such that on U

supp(P) ⊆ U_{0} × ⋯ × U_{N_{0}} × ∏_{N≥N_{0}} φ_{N_{0},N}(U_{N_{0}}),

where φ_{N_{0},N} = φ_{N} ∘⋯∘φ_{N_{0}} : U_{N_{0}} → U_{N+1}. We say that P has compact support.

Let 𝕍_{σ}^{K} be the category objects of which being couples (V, σ^{♯}), where V is a finite dimensional K-vector spaces and σ^{♯} an automorphism of V, then

Lemma 6.4. — Restricted on the category 𝔓_{c}, the functor ℍ(Gr, ·) takes values in 𝕍_{σ}^{K}.

Proposition 6.5. — Equipped with the symmetric product ⊛ and with the fiber functor ℍ(Gr, ·), the category 𝔓_{c} is a tannakian category.

7. The main result

Theorem 7.1. — Let A be the automorphism group of the fiber functor ℍ(Gr, ·) from 𝔓_{c} to 𝕍_{σ}^{K}, then A is isomorphic to the Langlands dual ^{L}G of G and we have an equivalence of categories

𝔓_{c} ≃ Rep_{σ}^{K}

between 𝔓_{c} and the category of finite dimensional K-representaton of ^{L}G compatible with the σ-actions.

The end of the theorem means that if ρ : ^{L}G → V is an object of Rep_{σ}^{K}, ρ commutes with the actions of σ on V and on ^{L}G, this last one being also the set of K-points of an algebraic linear group.

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