Geometric Realization of Whittaker Functions and the Langlands Conjecture

1.1. Let X be a smooth, complete, geometrically connected curve over Fq. Denote by F the field of rational functions on X , by A the ring of adeles of F , and by Gal(F/F ) the Galois group of F . The present paper may be considered as a step towards understanding the geometric Langlands correspondence between n–dimensional `–adic representations of Gal(F/F ) and automorphic forms on the group GLn(A). We follow the approach initiated by V. Drinfeld [7], who applied the theory of `–adic sheaves to establish this correspondence in the case of GL2. The Langlands conjecture predicts that to any unramified irreducible n-dimensional representation σ of Gal(F/F ), one can attach an unramified automorphic function fσ on GLn(A). The starting point of Drinfeld’s approach is the observation that an unramified automorphic function on the group GLn(A) can be viewed as a function on the set Mn of isomorphism classes of rank n bundles on the curve X . The set Mn is the set of Fq–points of Mn, the algebraic stack of rank n bundles on X . One may hope to construct the automorphic form associated to a Galois representation as a function corresponding to an `–adic perverse sheaf on Mn. This is essentially what Drinfeld did in [7] in the case of GL2. In abelian class field theory (the case of GL1) this was done previously by P. Deligne (see [22]).