Rigid vectors in $p$-adic principal series representations

Abstract

In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups $\mathbb{I}$ 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 $\mathbb{I}$-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.

21 pages