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Let (G,K) be a symmetric pair over the complex numbers, and let X=K\G be thecorresponding symmetric space. In this paper we study a nearby cycles functorassociated to a degeneration of X to MN\G, which we call the "wonderfuldegeneration". We show that on the category of character sheaves on X, thisfunctor is isomorphic to a composition of two averaging functors (a parallelresult, 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 functorof [ENV] and use this formula to give a geometric proof of the celebratedCasselman's submodule theorem and establish a second adjointness theorem forHarish-Chandra modules.

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