To Robert MacPherson on the occasion of his 60th birthday GEOMETRIC LANGLANDS CORRESPONDENCE FOR D-MODULES IN PRIME CHARACTERISTIC: THE GL(n) CASE

Let X be a smooth projective algebraic curve of genus > 1 over and algebraically closed field k of characteristic p > 0. Denote by Bunn (resp. Locn) the moduli stack of vector bundles of rank n on X (resp. the moduli stack of vector bundles of rank n endowed with a connection). Let also DBunn denote the sheaf of crystalline differential operators on Bunn (cf. e.g. [3]). In this paper we construct an equivalence Φn between the bounded derived category Db(M(OLoc0 n )) of quasi-coherent sheaves on some open subset Locn ⊂ Locn and the bounded derived category D (M(D Bunn)) of the category of modules over some localization D Bunn of DBunn . We show that this equivalence satisfies the Hecke eigen-value property in the manner predicted by the geometric Langlands conjecture. In particular, for any E ∈ Locn we construct a ”Hecke eigen-module” AutE . The main tools used in the construction are the Azumaya property of DBunn (cf. [3]) and the geometry of the Hitchin integrable system. The functor Φn is defined via a twisted version of the Fourier-Mukai transform.