On the irreducibility of $p$-adic Banach principal series of $p$-adic reductive groups

Abstract

Suppose that $G$ is the group of $F$-points of a connected reductive group over $F$, where $F/\mathbb{Q}_p$ is a finite extension. We study the (topological) irreducibility of principal series of $G$ on $p$-adic Banach spaces. For unitary inducing representations we obtain an optimal irreducibility criterion, and for $G = \mathrm{GL}_n(F)$ (as well as for arbitrary split groups under slightly stronger conditions) we obtain a variant of Schneider's conjecture [Sch06, Conjecture 2.5]. In general we reduce the irreducibility problem to smooth inducing representations and almost simple simply-connected $G$. Our methods include locally analytic representation theory, the bifunctor of Orlik--Strauch, translation functors, as well as new results on reducibility points of smooth parabolic inductions.