Geometric Langlands in prime characteristic

Abstract

Let G be a semi-simple algebraic group over an algebraically closed field k, whose characteristic is positive and does not divide the order of the Weyl group of G, and let Ğ be its Langlands dual group over k. Let C be a smooth projective curve over k of genus at least two. Denote by Bun_G the moduli stack of G-bundles on C and LocSys_Ğ the moduli stack of Ğ-local systems on C. Let D_(Bun_G) be the sheaf of crystalline differential operators on Bun_G. In this paper we construct an equivalence between the bounded derived category D^b (QCoh(LocSys^0_Ğ)) of quasi-coherent sheaves on some open subset LocSys^0_Ğ ⊂ LocSys_Ğ and bounded derived category D^b (D^0_(Bun_G) -mod) of modules over some localization D^0_(Bun_G) of D_(Bun_G). This generalizes the work of Bezrukavnikov and Braverman in the GL_n case.

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