Geometric Langlands in prime characteristic

Abstract

Let be a semi-simple algebraic group over an algebraically closed field , whose characteristic is positive and does not divide the order of the Weyl group of , and let be its Langlands dual group over . Let be a smooth projective curve over of genus at least two. Denote by the moduli stack of -bundles on and the moduli stack of -local systems on . Let be the sheaf of crystalline differential operators on . In this paper we construct an equivalence between the bounded derived category of quasi-coherent sheaves on some open subset and bounded derived category of modules over some localization of . This generalizes the work of Bezrukavnikov and Braverman in the case.