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.