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

Marko Tadic

Published 2016-01-27Version 1

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.