Geometric langlands correspondence for D-modules in prime characteristic: The GL(n) case

Abstract

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(OLocnO)) of quasi-coherent sheaves on some open subset LocnO ⊂ Locn and the bounded derived category Db(M(DBunnO)) of the category of modules over some localization DBunnO 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 ε ∈ LocnO we construct a "Hecke eigen-module" Autε. 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.