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

Let (G, K) be a symmetric pair over the complex numbers. We study a nearby cycles functor associated to a degeneration of the symmetric space K\G to the horospherical space MN\G. We show that on the category of admissible sheaves on K\G, this functor is isomorphic to a composition of two averaging functors. As an application, we give a geometric proof of Casselman’s submodule theorem. This is joint work with Tsao-Hsien Chen.