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

Noriyuki Abe; Florian Herzig

representation theory

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

arXiv:2303.13287

summary

The paper investigates when principal series representations of p‑adic reductive groups on Banach spaces are topologically irreducible, providing optimal criteria for unitary inductions and confirming a variant of Schneider’s conjecture for GLₙ and split groups.

Abstract

Suppose that $G$ is the group of $F$ -points of a connected reductive group over $F$ , where $F / Q_{p}$ 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 $G = GL_{n} (F)$ (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.

88 pages

Topics & keywords

#p-adic Banach representations #principal series #irreducibility criteria #locally analytic representations #reductive groupsprincipal seriesBanach spacep-adic groupslocally analyticSchneider's conjectureOrlik–Strauch bifunctortranslation functors

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

Abstract

Topics & keywords

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