International Congress of Mathematicians

Geometrie Langlands Correspondence for GL n 5732 we define Hecke eigen-sheaves and state the main theorem about the existence ofthe Hecke eigen-sheaf JE corresponding to an irreducible local system E. In Section3 we describe the construction of JE via a geometric analog of the Whittaker model.In Section 4 we explain how the main theorem about the existence of JE followsfrom a certain vanishing result. Finally, in Section 5 we indicate the steps involvedin the proof of the vanishing theorem.1. The classical theoryIn this section we will review the formulation of the classical Langlands conjecturefor function fields, and the technique of construction of automorphic formsvia Whittaker models.1.1. Let X be the global field corresponding to a (smooth, complete) curve X overa finite field ¥ q . We will denote by  (resp., O) the corresponding ring of adeles(resp., the subring of integral adeles).Consider the quotient GL n (X)\GL n (K). The space Funct(GL n (X)\GL n (Âj)of (smooth) functions (with values in an arbitrary algebraically closed field of char.0, which we will take to be Q f ) is naturally a representation of the group GL n (A).Consider the subspace of functions that are invariant with respect to GL n (Q),i.e. the space of functions on the double quotient GL n (X)\GL n (K)/GL n (&). Thisis a module over the Hecke algebra GL n (K) with respect to GL n (