p-adic boundary values

TL;DR

This paper develops boundary value maps for p-adic group representations, characterizing their images as functions satisfying specific transformation properties and differential equations, extending prior work on p-adic symmetric spaces.

Contribution

It constructs and characterizes boundary value maps for p-adic group representations, generalizing earlier results and extending the theory of such representations.

Findings

Boundary value maps are constructed for p-adic representations.

Images of these maps are characterized by transformation properties and hypergeometric PDEs.

The work generalizes Morita's and Schneider-Stuhler's results on p-adic groups and symmetric spaces.

Abstract

We study in detail certain natural continuous representations of G = GL(n,K) in locally convex vector spaces over a locally compact, non-archimedean field K of characteristic zero. We construct boundary value maps, or integral transforms, between subquotients of the dual of a ``holomorphic'' representation coming from a p-adic symmetric space, and ``principal series'' representations constructed from locally analytic functions on G. We characterize the image of each of our integral transforms as a space of functions on $G$ having certain transformation properties and satisfying a system of partial differential equations of hypergeometric type. This work generalizes earlier work of Morita, who studied this type of representation of the group SL(2,K). It also extends the work of Schneider-Stuhler on the deRham cohomology of p-adic symmetric spaces. We view this work as part of a general program of developing the theory of such representations.