Remark on representation theory of general linear groups over a non-archimedean local division algebra

TL;DR

This paper provides a straightforward local proof of key results in the representation theory of general linear groups over non-archimedean local division algebras, focusing on tempered and square integrable representations.

Contribution

It offers a simple, local proof of the parameterization of irreducible square integrable representations and the irreducibility of tempered parabolic induction for these groups.

Findings

Parameterization of irreducible square integrable representations by segments of cuspidal representations

Proof of irreducibility of tempered parabolic induction

Utilization of basic facts and classical results in representation theory

Abstract

In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorporated in the structure of a Hopf algebra). We use only some very basic general facts of the representation theory of reductive p-adic groups (the theory that we use was completed more then three decades ago, mainly in 1970-es). Of the specific results for general linear groups over A, basically we use only a very old result of G.I. Olshanskii, which says that there exist complementary series starting from $Ind(\rho\otimes\rho)$ whenever $\rho$ is a unitary irreducible cuspidal representation. In appendix of the paper "On parabolic induction on inner forms of the general linear group over a non-archimedean local field" of E. Lapid and A. Minguez, there is also a simple local proof of these results, based on a slightly different approach.