Geometric Langlands in prime characteristic

Let $G$ be a semisimple 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 $\breve G$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$. Denote by \Bun_G the moduli stack of $G$-bundles on $C$ and \Loc_{\breve G} the moduli stack of $\breve G$-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(\on{QCoh}(\Loc_{\breve G}^0)) of quasi-coherent sheaves on some open subset \Loc_{\breve G}^0\subset\Loc_{\breve G} and bounded derived category D^b(D_{\Bun_G}^0\on{-mod}) of modules over some localization D_{\Bun_G}^0 of D_{\Bun_G}. This generalizes the work of Bezrukavnikov-Braverman in the \GL_n case.Comment: 57 pages, corrected some arguments in section 3.6 and 3.7, to appear in Compositio Mat

Affine Matsuki correspondence for sheaves

We lift the affine Matsuki correspondence between real and symmetric loop group orbits in affine Grassmannians to an equivalence of derived categories of sheaves. In analogy with the finite-dimensional setting, our arguments depend upon the Morse theory of energy functions obtained from symmetrizations of coadjoint orbits. The additional fusion structures of the affine setting lead to further equivalences with Schubert constructible derived categories of sheaves on real affine Grassmannians

Non-abelian Hodge theory for algebraic curves in characteristic p

Let G be a reductive group over an algebraically closed field of positive characteristic. Let C be a smooth projective curve over k. We give a description of the moduli space of flat G-bundles in terms of the moduli space of G-Higgs bundles over the Frobenius twist C' of C. This description can be regarded as the non-abelian Hodge theory for curves in positive characteristic.Comment: The introduction and the example for GL_n are partially rewritten. Section 3 is re-organized. Various typos are corrected. To appear in GAF

A Formula for the Geometric Jacquet Functor and its Character Sheaf Analogue

Let (G,K) be a symmetric pair over the complex numbers, and let X=K\G be the corresponding symmetric space. In this paper we study a nearby cycles functor associated to a degeneration of X to MN\G, which we call the "wonderful degeneration". We show that on the category of character sheaves on X, this functor is isomorphic to a composition of two averaging functors (a parallel result, on the level of functions in the p-adic setting, was obtained in [BK, SV]). As an application, we obtain a formula for the geometric Jacquet functor of [ENV] and use this formula to give a geometric proof of the celebrated Casselman's submodule theorem and establish a second adjointness theorem for Harish-Chandra modules.Comment: Revised version. Equivariancy replaces stratification arguments, so that the results are applicable to all sheaf setting