Rigid vectors in $p$-adic principal series representations

TL;DR

This paper generalizes the theory of pro-$p$ Iwahori subgroups as rigid analytic groups for large $p$, and provides an irreducibility criterion for principal series representations of split reductive groups over $p$-adic fields.

Contribution

It extends Lazard's results to a broader class of groups and establishes a new irreducibility criterion for principal series representations.

Findings

Pro-$p$ Iwahori subgroups can be viewed as rigid analytic groups for large $p$.

The paper generalizes Lazard's results from $ ext{GL}_n$ to arbitrary split reductive groups.

An irreducibility criterion for principal series representations is established.

Abstract

In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups $\Bbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural $p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_n$. We work with a general connected reductive split group over some $p$-adic field (with simply connected derived group) and study the $\Bbb{I}$-analytic vectors in principal series representations. Our main result is an irreducibility criterion which generalizes results of Clozel and Ray in the $\text{GL}_n$-case.