Hitchin fibration and endoscopy

Transcription

1 Hitchin fibration and endoscopy Talk given in Kyoto the 2nd of September 2004 In [Hitchin-Duke], N. Hitchin proved that the cotangent of the moduli space of G-bundle over a compact Riemann surface is naturally a completely integrable system. In this talk, I will show how the endoscopy theory can be very naturally interpreted in terms of the cohomology of the Hitchin fibration. Hitchin pairs. Let k = F q be a finite field, k its algebraic closure. Let X be a geometrically connected smooth projective curve over k. Let denote F = k(x) the field of rational functions over X. Let X denote the set of closed points of X. We are going to fix an effective divisor D = v X d v[v] of large degree. Let G be a group scheme over X whose geometric fibers are connected reductive groups. One can define the notion of Hitchin pairs associated to a triple (X,G,D). If G = GL n, a Hitchin pair consists in a pair (E,ϕ) where E is a vector bundle of X and ϕ is a twisted endomorphism of E ϕ : E E(D) := E OX O X (D). In the general case, a Hitchin pair consists in a pair (E,ϕ) where E is a G-bundle over X, and where ϕ is a section of the vector bundle ϕ H 0 (X,ad(E)(D)). Here ad(e) is the vector bundle obtained by pushing out the torsor E be the adjoint representation of G. For any test k-scheme, we can generalized the notion Hitchin pair to Hitchin pair parameterized by S in the straitforward way. This allows us to define a k-stack M that associates to any k-scheme S the category of Hitchin pairs parameterized by S. This is an algebraic stack since the moduli of G-bundles is an algebraic stack. 1

2 Counting. Just as the moduli of vector bundle, the algebraic stack M is not of finite type, we can nevertheless try to describe the k-points of M in adelic terms, like the way Weil counts the number of vector bundles. Let (E,φ) be a k-point of M. The isomorphism class of the generic fiber E η of E defines an element cl F (E) H 1 (F,G). For every place v F, the isomorphism class cl Fv (E) H 1 (F v,g) comes from an element of H 1 (O v,g) which is trivial by a theorem of Lang. Therefore cl F (E) ker 1 (F,G) := ker[h 1 (F,G) v H 1 (F v,g)]. This set ker 1 (F,G) is in general a small set which is just a singleton if G is a semi-simple simply connected or adjoint group. For simplicity, we will assume that ker 1 (F,G) is trivial. Lets choose a trivialization ι of the generic fiber E F of E. The difference between E and the trivial G-bundle can be encoded in the data (g v ) v F with g v G(F v )/G(O v ) which are the trivial classes for almost all places v. Using the trivialization ι, a rational section ϕ H 0 (F,ad(E)) is given by an element γ g(f ). The necessary and sufficient condition for γ to be extended to a integral section ϕ H 0 (X,ad(E)(D)) is that for all places v F, we have It follows the formal expression g v γg 1 v ϖ dv g(o v ). M(k) = γ vol(g γ (F )\G γ (A))O γ (1 D ) where we sum over a set of representatives of G(F )-conjugacy classes in g(f ) and where the characteristic function 1 D of the divisor D is 1 D = v F 1 ϕ dv g(o v) and where the orbital integral O γ (1 D ) is the adelic orbital integral O γ (1 D ) = 1 D (g 1 γg). G γ (A)\G(A) The term vol(g γ (F )\G γ (A)) can be infinite but we will restrict ourself to elliptic elements γ of semi-simple groups G for which this volume is finite. In any case, this formal expression will be an excellent guide for us. 2

3 Hitchin fibration. Let us consider the GL n case first. Let (E,φ) be a Hitchin pair for GL n where E is a vector bundle of rank n on X and ϕ is a twisted endomorphisme ϕ : E E(D). Consider the exterior product and its trace i ϕ : i E i E(iD) tr( i ϕ) H 0 (X,O X (id)). This defines the Hitchin fibration f : M A from M to the affine space A = n H 0 (X,O X (id)). i=1 We will define the Hitchin fibration in the case G comes from a reductive over k. Recall the theorem of Chevalley : k[g] G = k[t] W = k[u 1,...,u r ]. The subalgebra of ad(g)-invariant regular functions on g is a polynomial algebra : the generators u 1,...,u r are homogenous polynomial of degree deg(u i ) = m i. There is no really good way to choose these homogenous polynomials, but their degrees deg(u i ) = m i are independent of any choice and are called the exponents of G. The affine space car = Spec(k[u 1,...,u r ]) is equipped with the natural G m action given by t(u 1,...,u r ) = (t m 1 u 1,...,t mr u r ). With respect to this action the Chevalley map χ : g car is G-equivariant with respect to the adjoint action of G on g and the trivial action on car, and is G m -equivariant with respect to the homothety on g and to G m -action on car that is just defined above. Let denote [χ] : [g/g G m ] [car/g m ] the stacky map deduced from χ. We need the following tautological observation. Lemma 1 Let X be any scheme and let G be any algebraic group. A pair (E,θ) where E is a G-torsor on X and θ is a section of the vector bundle 3

4 obtained by pushing out E by a representation G GL(V ) is equivalent to a map h E,θ : X [V/G] that lifts the map h E : X BG to the classifying space of G corresponding to the G-torsor E. Thus the Hitchin pair (E,ϕ) correspond to a map h E,ϕ : X [g/g G m ] that lifts the map X BG BG m corresponding to the G-torsor E and to the G m -torsor L D on X associated to the O X -invertible sheaf O X (D). Consider the following diagram h E,ϕ h a X [g/(g G m )] [χ] [car/g m )] h D h E h D B(G G m ) BG m The Hitchin pair defines the map h E,φ. By composing h E,φ with [χ] we get a map h a : X [car/g m ] corresponding to a global section a H 0 (X,car G m L D ) = A of the vector bundle obtained by pushing out the G m -torsor L D by the representation of G m on car. Thus we get the Hitchin map f : M A. For all a A(k), by repeating the counting the number of points of the fiber M a = f 1 (a) of the Hitchin map, we get the formal expression M a (k) = vol(g γ (F )\G γ (A))O γ (1 D ) χ(γ)=a where we sum over a set of representatives of G(F )-conjugacy classes in g(f ) whose image in car(f ) is a H 0 (X,car G m L D ) car(f ). This expression makes sense for elliptic a. What we want is to obtain a decomposition of the direct image f Q l that corresponds to the decomposition of the trace formula into κ-parts. More precisely, we will construct a canonical decomposition of the perverse cohomology sheaves p H j (f Q l ) over the open subvariety of A corresponding 4

5 to elliptic a. This decomposition will derive from the action of a group scheme over A, more precisely, an action of a relative Picard stack on the fibers of f. Picard stack P a. The appearance of the factor vol(g γ (F )\G γ (A)) in the expression of M a, suggests there should be an action of some kind of group on M a. In reality, there will be a natural action of a Picard stack P a on the fiber M a. To define P a, we will need a few preparation. The I be the group scheme over g of centralizers which is not a flat over g. I = {(x,g) g G gxg 1 = x} Lemma 2 There exists a unique smooth groups scheme J over g equipped with a homomorphism χ J I which is an isomorphisme over the open set g reg of regular elements of g. The restriction of I to g reg is a smooth commutative group scheme. Two regular elements γ,γ such that χ(γ) = χ(γ ) are necessarily conjugate. Moreover, as we have already observed in the first talk, since Iγ is commutative, the conjugation defines a canonical isomorphism I γ = I γ. Using faithfully flat descent, we have a smooth group scheme J over car such that χ J g reg I g reg. Now χ J is smooth so normal, I is an affine scheme, g g reg is of codimension more than 2, this isomorphism extends to a unique map χ J I which is necessarily a homomorphism of groups. In the GL n case, this construction is quite familiar. Let x be an endomorphism of V = k n whose the caracteristic polynomlial is a moni polynomial a k[t]. Then we have a map k[t]/(a) End(V,x) sending x on t whose image is k[t]/(b) where b is the minimal polynomial of x. This induces a homomorphisme of group J a = (k[t]/(a)) End(V,x) = I x. The smooth commutative group scheme J over car is obviously equivariant under the G m -action on car. This makes it possible to descent J to a smooth group scheme [J] over car/g m. Let a A a point of the Hitchin affine space, h a : X [car/g m ] the associated map. Let J a = h a[j] and let P a be the Picard stack of J a -torsors over X. Action of P a on M a. We claim that there exists a natural action of P a on M a. Let (E,ϕ) be a point of M a. Its group scheme of automorphisms 5

6 over X is I E,ϕ = h E,ϕ [I]. Thus we have a homomorphism J a = h a[j] I E,ϕ derived from the homomorphism χ J I. Therefore we can twist (E,ϕ) by any J a -torsor. Let us illustrate this quite abstract construction in the familiar case of GL n. Following Hitchin, Beauville-Narasimhan-Ramanan, for GL n, one can associate to a A a spectral covering Y a X which is a finite covering of degree n. The fiber M a is the compactified Jacobian of Y a M a = { torsion-free O Ya -Module of generic rank 1} P a is the Jacobian of Y a P a = { invertible O Ya -Module} and P a acts on M a by tensor product. Sheaf of connected components of P a. Let g ssr be the open subvariety of g consisting in elements which are semi-simple regular. There exists an open subvariety car ssr car such that χ 1 (car ssr ) = g ssr. Let A be the open subvariety of a consisting of elements a A such that the map h a : X [car/g m ] send the generic point of X into the open [car ssr /G m ]. Lemma 3 The relative Picard stack P/A is smooth over the open A. Thus, by a theorem in EGA IV.3, there exists a unique open substack P 0 of P such that for all geometric point a of A, Pa 0 is the neutral connected component of P a. By taking the quotient, we obtain a sheaf π 0 (P/A ) for the etale topology of A such that for all geometric point a A of A, the fiber of π 0 (P/A ) over a is π 0 (P/A ) a = π 0 (P a ). The action of the relative Picard stack P/A on M/A induces an action of P on the direct image f Q l A. By the homotopy lemma, the action of the Picard stack of neutral components P 0 is trivial and therefore we get an action of the sheaf π 0 (P/A ) on the perverse sheaves p H j (f Q l ) A. Let assume G to be a semi-simple group. Let A ell be the open subvariety of A where the sheaf π 0 (P/A ell ) is finite. Over this open subvariety, the action of π 0 (P/A ell ) on p H j (f Q l ) A ell leads to a decomposition in direct sum by the following general formalism. Co-sheaf of characters. Let X be a scheme, F by a sheaf for the etale topology of X and let K be a perverse sheaf on X. Assume that F acts on 6

7 K that means for all U/X, we are given an action of F(U) on K U which is compatible with respect to different U. For every U, let F(U) to be the group of characters of the finite abelian group F(U). The action of F(U) on K U induces a decomposition into direct sum K = χ F(U) K χ. Let consider the covariant functor U F(U) which associates to an etale map [U U] the map on characters groups [F(U ) F(U) ] which is a precosheaf. We can associate to it a cosheaf of sets F co equipped with a morphism of functors F co (U) F(U) and verifying the co-sheaf condition : for all etale map U/X, for all etale covering U 1 /U, U 2 = U 1 U U 1, we have an exact sequence F co (U 2 ) F co (U 1 ) F co (U) and which is universal for these properties. We have a decomposition K = χ F co (X) where F co (X) is not necessarily an abelian group anymore. Let us consider two typical examples. Let x be a geometric point of X and let F be the finite locally constant sheaf given by an action of Γ = π 1 (X,x) on a finite abelian group A. Then the set of global section of the co-sheaf F co is the set of co-invariants A Γ. If F acts on a perverse sheaf K then locally we have a decomposition K = χ A K χ. Globally, only the Γ-orbits of χ is well defined and therefore we have a decomposition K = [χ] A Γ K [χ]. This example illustrates how one can passes from the decomposition to κ-orbital integral in a neighborhood of an element to a coarser decomposition for the whole elliptic part of the trace formula. The second example is as follows : let X be the affine line and let F be the skyscraper sheaf Z/2Z supported by the origine. In this case F co (U) = Z/2Z if 0 U and equal to 0 if 0 / U. Let F act on a perverse sheaf K, then we get a decomposition K = K + K. The following wimple observation is going to be crucial : over U = X {0}, F is the trivial sheaf so that K U = K + U, thus K is supported by {0}. Assume that K is a perverse sheaf then the minus part of its fiber over 0 is pure. This assertion is quite similar to the purity conjecture formulated by Goresky-Kottwitz-MacPherson which is an important ingredient in their approach to the fundamental lemma using equivariant cohomology. Description of π 0 (P a ). We need a more detailed description of π 0 (P/A ell ) in order make the decomposition of p H j (f K) more explicite. Let a be a 7 K χ

8 geometric point of A and let h a : X [car/g m ] be the associated map. Let U a be the preimage of [car ssr /G m ]. The -condition on a means that U a is not empty. The restriction to U a of the smooth group scheme J a = h [J] over X, is a torus. The fibers of J a over x X U a can have additive part, and can even be non-connected. From now on, I will assume that G is an adjoint semi-simple group. In this case, J a is a smooth group scheme with connected fibers. Lemma 4 Let G be an adjoint semi-simple group. Let a be a geometric point of A. Let U a be the open subvariety of X where J a is a torus, let X a be the sheaf of cocharacters of this torus which is a locally constant sheaf of free abelian groups. Let u be a geometric point of U a then π 0 (P a ) is the maximal π 1 (U a,u)-invariant quotient of the fiber (X a) u. The fiber π 0 (P a ) depends on the monodromy representation of the cocharacter sheaf which is given by a homomorphism ρ(a,u) : π 1 (U a,u) W where W is the Weyl group whose image is a subgroup Σ(a,u) of W. This subgroup depends on the choice of the base point u but its conjugacy class [Σ(a)] does not. A point a is elliptic if and only if there is no Σ(a,u)-invariant in X a. We introduce the stratification A = [Σ] A [Σ] where a A [Σ] if and only if [Σ(a)] = [Σ]. The constructible sheaf π 0 (P/A ) is locally constant along the strata A [Σ] under the assumption that G is an adjoint semi-simple group. We say that a subgroup Σ W is elliptic if the group of Σ-coinvariants X is finite. We have A ell = [Σ] A Σ union over the conjugacy classes of elliptic subgroups Σ W. The fiber of the co-sheaf π 0 (P/A ell ) over a geometric point a is π 0 (P/A ell ) a = {π 1 (U a,u)-invariant characters of X a}. Thus we recover the familiar group ˆT Γ where ˆT is the maximal torus if the dual group Ĝ equipped with the action of Γ coming form the centralizer G γ 8

9 of any element γ g(f ) whose the characteristic polynomial is a. One can check that the set of sections of the co-sheaf π 0 (P/A ) co is {κ ˆT Fix W (κ) is an elliptic subgroup}/w. Let κ ˆT such that the fixator subgroup Fix(κ) is an elliptic subgroup. Since G is assumed to be a semi-simple adjoint group, its dual Ĝ is a semisimple simply connected group. Therefore, the centralizer Ĝκ is a connected reductive group. The group Ĥ = Ĝκ. is generated by ˆT and the roots that are trivial on κ. The Weyl group W H of Ĥ is just the fixator group of κ in W. Let H be the split reductive over X whose dual group os Ĥ (for adjoint semi-simple groups, we only have split endoscopic groups). We have a natural map T/W H = car H T/W = car which induces a natural map A H A from the Hitchin affine space A H of H to that of G. Lemma 5 The image of the map A H A is the closure A [Σ] of the stratum A [Σ] with Σ = W H. Over the open stratum, the map A H A[WH ] A [W H ] A [WH ] is a finite etale map of degree Nor W (W H ) / W H. Theorem 6 Assume that G is a semi-simple adjoint group 1. The action of π 0 (P/A ell ) over K = p H j (f Q l A ell) induces a canonocal decomposition of this perverse sheaf K = [κ] K [κ] where direct sum runs over the set of W -conjugacy classes [κ] of κ ˆT such that the fixator Fix W (κ) of κ in W is elliptic. Moreover, for every such κ, let H be the associated elliptic endoscopic group. Then the perverse sheaf K [κ] is supported by the image of the map A H A. We can reformulate of the stabilization conjecture geometrically in somehow vague terms : Conjecture 7 The pullback of K κ to A H A[WH A [WH ] is a pure perverse ] sheaf, is isomorphic to the κ = 1 part of the perverse cohomology of the Hitchin map of H, up to a shift and and a twist by a finite local system of rank one and of order two. 1. The theorem is now proved without the assumption G adjoint 9

10 The shift and the twist correspond to the power of q and the sign in the Langlands-Shelstad transfer factor. This conjecture has been proved by Laumon and myself in the case un unitary group. In that case, the construction of the local system corresponding to sign is quite subtle. Purity statement. The purity statement in the conjecture is an immediate consequence of the theorem. Indeed, the map f : M A is a proper map over A ell. The total space of M over A ell is smooth. Consequently, f Q l A ell is a pure complex. Their perverse cohomology K = p H j (f Q l A ell) are pure perverse sheaves. The purity of the pullback of K κ to A H A[WH ] A [W H ] derives from the fact that K κ is supported by the image of A H A, and this map is finite etale over the stratum A [WH ]. 10