Resolutions for principal series representations of p-adic GL(n)

TL;DR

This paper extends the Schneider-Stuhler resolution method to principal series representations of GL(n,F) over fields of arbitrary characteristic, providing a new way to realize these representations as 0-homology of coefficient systems.

Contribution

It demonstrates that principal series representations of GL(n,F) over any characteristic field can be realized as 0-homology, overcoming previous limitations for mod p representations.

Findings

Principal series representations can be realized as 0-homology.

Method applies to fields of arbitrary characteristic.

Extends Schneider-Stuhler resolution to new cases.

Abstract

Let F be a nonarchimedean locally compact field with residue characteristic p and G(F) the group of F-rational points of a connected reductive group. Following Schneider and Stuhler, one can realize, in a functorial way, any smooth complex finitely generated representation of G(F) as the 0-homology of a certain coefficient system on the semi-simple building of G(F). It is known that this method does not apply in general for smooth mod p representations of G(F), even when G= GL(2). However, we prove that a principal series representation of GL(n,F) over a field with arbitrary characteristic can be realized as the 0-homology of the corresponding coefficient system.