[Paper Review] p-adic boundary values

Peter Schneider, Jeremy Teitelbaum|ArXiv.org|Jan 28, 1999

Advanced Algebra and Geometry16 references3 citations

TL;DR

This paper constructs p-adic boundary value maps—integral transforms—between duals of holomorphic representations on p-adic symmetric spaces and principal series representations built from locally analytic functions on GL(n,K). It characterizes the image of these transforms as spaces of functions on G satisfying specific transformation properties and a system of hypergeometric-type partial differential equations, generalizing Morita's work on SL(2,K) and extending Schneider-Stuhler's de Rham cohomology results in the p-adic setting.

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.

Motivation & Objective

• To develop a general theory of continuous representations of GL(n,K) in locally convex vector spaces over non-archimedean fields.
• To construct integral transforms (boundary value maps) between subquotients of duals of holomorphic representations and principal series representations.
• To characterize the image of these transforms as spaces of functions on G with specific transformation properties and hypergeometric-type PDEs.
• To generalize Morita's earlier results on SL(2,K) to the general linear group GL(n,K).
• To extend the framework of Schneider-Stuhler's de Rham cohomology of p-adic symmetric spaces to include boundary value theory.

Proposed method

• Constructs representations of GL(n,K) using locally convex vector spaces over a non-archimedean field K of characteristic zero.
• Defines holomorphic representations via p-adic symmetric spaces and considers their duals.
• Introduces boundary value maps as integral transforms from duals of holomorphic representations to principal series representations.
• Uses locally analytic functions on G to build the principal series representations.
• Analyzes the image of the transforms by deriving a system of partial differential equations of hypergeometric type.
• Applies techniques from p-adic analysis and representation theory to characterize the image spaces.

Experimental results

Research questions

• RQ1 What is the structure of the image of the p-adic boundary value maps from holomorphic representations to principal series representations on GL(n,K)?
• RQ2 How do the transformation properties of the image functions relate to the group action on G?
• RQ3 What system of partial differential equations characterizes the image of the integral transform in the p-adic setting?
• RQ4 In what way does this construction generalize Morita's work on SL(2,K) to GL(n,K)?
• RQ5 How does this boundary value theory extend the de Rham cohomology framework of Schneider-Stuhler for p-adic symmetric spaces?

Key findings

• The image of each boundary value map is precisely characterized as a space of functions on G that satisfy specific transformation properties under the group action.
• These image functions are shown to satisfy a system of partial differential equations of hypergeometric type.
• The construction generalizes Morita's results on SL(2,K) to the higher-rank case GL(n,K).
• The boundary value maps provide a bridge between holomorphic representations on p-adic symmetric spaces and principal series representations.
• The results extend the de Rham cohomology theory of p-adic symmetric spaces by incorporating boundary value structures.
• The method establishes a new framework for studying p-adic representations via integral transforms and PDEs.