A formula for the geometric Jacquet functor and its character sheaf analogue

Creators: Chen, Tsao-Hsien; Yom Din, Alexander

Abstract

Let (G,K) be a symmetric pair over the complex numbers, and let X=K∖GX=K∖G be the corresponding symmetric space. In this paper we study a nearby cycles functor associated to a degeneration of X to MN∖GMN∖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.

Additional Information

© 2017 Springer International Publishing AG. Received: October 11, 2016; Revised: May 17, 2017; Accepted: May 23, 2017. A.Y.D. would like to thank his PhD advisor Joseph Bernstein, for suggesting to study the Casselman-Jacquet functor algebraically. Both authors would like to thank D. Gaitsgory for very helpful comments. Both authors would like to thank the Hausdorff Research Institute for Mathematics and Max Planck Institute for Mathematics, for excellent hosting and working conditions in the summer of 2014, during which the cooperation began. T.H.C. was partially supported by an AMS-Simons Travel Grant. A.Y.D. was partially supported by ERC grant 291612 and by ISF grant 533/14.

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