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 M N \ 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.