On the Geometric Langlands Conjecture

0.1. Background. Let X be a smooth, complete, geometrically connected curve over the finite field Fq. Denote by F the field of rational functions on X and by A the ring of adèles of F . The Langlands conjecture, recently proved by L. Lafforgue [Laf], establishes a correspondence between cuspidal automorphic forms on the group GLn(A) and irreducible, almost everywhere unramified, n–dimensional `– adic representations of the Galois group of F over F (more precisely, of the Weil group). An unramified automorphic form on the group GLn(A) can be viewed as a function on the set Bunn(Fq) of isomorphism classes of rank n bundles on the curve X . The set Bunn(Fq) is the set of Fq–points of Bunn, the algebraic stack of rank n bundles on X . According to Grothendieck’s “faisceaux–fonctions” correspondence, one can attach to an `–adic perverse sheaf on Bunn a function on Bunn(Fq) by taking the traces of the Frobenius on the stalks. V. Drinfeld’s geometric proof [Dr] of the Langlands conjecture for GL2 (and earlier geometric interpretation of the abelian class field theory by P. Deligne, see [Lau1]) opened the possibility that automorphic forms may be constructed as the functions associated to perverse sheaves on Bunn. Thus, one is led to a geometric version of the Langlands conjecture proposed by V. Drinfeld and G. Laumon: for each geometrically irreducible rank n local system E on X there exists a perverse sheaf AutE on Bunn (irreducible on each component), which is a Hecke eigensheaf with respect to E, in an appropriate sense (see [Lau1] or Sect. 1 below for the precise formulation). Moreover, the geometric Langlands conjecture can be made over an arbitrary field k. Building on the ideas of Drinfeld’s work [Dr], G. Laumon gave a conjectural construction of AutE in [Lau1, Lau2]. More precisely, he attached to each rank n local system E on X a complex of perverse sheaves AutE on the moduli stack Bun ′ n of pairs {M, s}, where M ∈ Bunn is a rank n bundle on X and s is a regular nonzero section of M. He conjectured that if E is geometrically irreducible, then this sheaf descends to a perverse sheaf AutE on Bunn (irreducible on each component), which is a Hecke eigensheaf with respect to E. In our previous work [FGKV], joint with D. Kazhdan, we have shown that the function on Bunn(Fq) associated to Aut ′ E agrees with the function constructed previously by I.I. Piatetskii-Shapiro [PS1] and J.A. Shalika [Sha], as anticipated by Laumon [Lau2]. This provided a consistency check for Laumon’s construction.