Resolutions of locally analytic principal series representations of $GL_2$

TL;DR

This paper constructs a chain complex resolution for locally analytic principal series representations of GL_2 over a p-adic field, extending Schneider and Stuhler's work from smooth to locally analytic representations.

Contribution

It introduces a new coefficient system on the Bruhat-Tits tree for locally analytic representations, providing explicit resolutions analogous to the smooth case.

Findings

The chain complex is a resolution of the representation V.

The construction generalizes existing smooth representation resolutions.

Provides tools for analyzing locally analytic principal series representations.

Abstract

For a finite field extension $F/\mathbb{Q}_p$ we associate a coefficient system attached on the Bruhat-Tits tree of $G:= {\rm GL}_2(F)$ to a locally analytic representation $V$ of $G$. This is done in analogy to the work of Schneider and Stuhler for smooth representations. This coefficient system furnishes a chain-complex which is shown, in the case of locally analytic principal series representations $V$, to be a resolution of $V$.