Resolutions of tempered representations of reductive p-adic groups

TL;DR

This paper compares Ext-groups of tempered representations of reductive p-adic groups across various module categories, simplifying existing proofs using Bruhat-Tits buildings and Schneider-Stuhler resolutions.

Contribution

It provides a unified comparison of Ext-groups in different module categories and simplifies proofs of existing theorems using geometric and analytic methods.

Findings

Comparison of Ext-groups across module categories

Simplified proofs of known theorems

Use of Bruhat-Tits building and Schneider-Stuhler resolutions

Abstract

Let G be a reductive group over a non-archimedean local field and let S(G) be its Schwartz algebra. We compare Ext-groups of tempered G-representations in several module categories: smooth G-representations, algebraic S(G)-modules, bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces which contains all admissible S(G)-modules. We simplify the proofs of known comparison theorems for these Ext-groups, due to Meyer and Schneider-Zink. Our method is based on the Bruhat-Tits building of G and on analytic properties of the Schneider-Stuhler resolutions.