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Let ( G, K) be a symmetric pair over the complex numbers, and let $${X=K \backslash G}$$ be the corresponding symmetric space. In this paper we study a nearby cycles functor associated to a degeneration of X to $${MN \backslash G}$$ , 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]