International Congress of Mathematicians

ICM 2002 • Vol. II • 571-582Geometrie LanglandsCorrespondence for GL nD. Gaitsgory*AbstractWe will review the geometric Langlands theory (mainly for the groupGLn), whose development was initiated in the works of V. Drinfeld and G.Laumon.Let A be a (smooth, complete) algebraic curve over a ground field fc, andlet E be an ^-adic n-dimensional irreducible local system on X.The geometric Langlands conjecture (for GL„) says that to E one canassociate an automorphic sheaf SE, which is a perverse sheaf on the modulistack Bun n (A) classifying vector bundles of rank noni.We will explain the motivation for this conjecture in terms of the classicaltheory of automorphic forms, and the methods involved in the constructionof SE-2000 Mathematics Subject Classification: 14H60.Introduction0.1. Let X be a (smooth, complete) curve over a ground field fc, and G a (split)reductive group. In the main body of the paper we will assume that G = GL n ,but now we would like to make a general overview of the theory, in which G can bearbitrary. Let Bunc denote the moduli stack of G-bundles on X.Our object of study is the category D(Bunc)-the derived category of sheaveson Bunc with constructible cohomology. By "sheaves" we will mean either Q r adicsheaves, which can be done over any fc, or D-modules, when char(fc) = 0.Finally, let G be the Langlands dual of G (thought of as an algebraic groupover Q f or fc, depending on the sheaf-theoretic context).0.2. It is believed that if o is a G-local system on X (i.e., a homomorphism fromthe appropriate version of ni(X) to G), which is sufficiently generic, then to o onecan attach an automotphic sheaf 3> £ D(Bunc), which is a Hecke eigen-sheaf withrespect to a. (See [BD], Section 5 for the definition of the Hecke eigen-property foran arbitrary G, or Section 2 below for the GL n case.)* Department of Mathematics, The University of Chicago, 5734 University Ave., Chicago, IL60637, USA. E-mail: gaitsgde@math.uchicago.edu