The geometric correspondence in some special cases

The geometric Langlands correspondence for function fields over finite fields has been proved by Frenkel, Gaitsgory, Vilonen in [FGV]. The aim of this article is to write translation for curves over C and prove the correspondence in some special cases and some new cases, outside the frame of the usual geometric Langlands correspondence. 1 The geometric version of Langlands conjecture We start by recalling the geometric Langlands conjecture expressed in terms of Hecke operators. We refer to the book ’introduction to the Langlands program’ [BG] for this first part. We consider a projective smooth curve X/Fq of genus g > 0 which is geometrically irreducible, that is to say such that X×Fq Fq is irreducible. Note that we write X×Fq Fq instead of X×Spec(Fq) Spec(Fq). The stack of rank n vector bundles is denoted by Bunn,X and we denote by Bunn,X(Fq) (or simply Bunn(Fq)) the set of its Fq-points. Recall that an Fq-point of the stack Bunn,X is a morphism of stacks Spec(Fq) → Bunn,X Such a morphism is determined by the image of the identity map which is an object of Bunn,X over X × Spec(Fq) therefore an Fq-point of Bunn,X can be identified with a vector bundle of rank n over X . For i = 1 . . . n, one can define a stack Heckei over SchC such that for any scheme T over Fq Heckei(T ) denote the set of tuples (M,M, x), where x is closed point of X and M,M are vector bundles of rank n over X ×Fq T satisfying M ′ ⊂ M and M / M ≃ ( OX×FqT / OX×FqT (−[x× T ]) )i . (Cf [Lau] or [Fre]) The i-th Hecke correspondence is given by: Heckei supp×h && N N N N N N N N N N N h yyss ss ss ss s Bunn,X X × Bunn,X where h(M,M, x) = (M), h(M,M, x) = M and supp(M,M, x) = x. Let x ∈ |X | and write Heckei,x = supp (x). This gives a correspondence between {x} × Bunn,X and Bunn,X . We can show that the double quotient GLn(F ) ∖ GLn(A) / GLn(O) is in bijection with the set of (equivalence classes of) Fq-points of Bunn,X . Therefore the diagram defines an operator on the space of functions on GLn(F ) ∖ GLn(A) / GLn(O) which associates to f the function h→! ((h )f). The operator h→! is the integration of the function along the fibres of h . We can check that this operator is precisely the i-th Hecke operator Hi,x. The (perverse) sheaves corresponding to the automorphic functions associated to the automorphic representations (which occur in the ordinary