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

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. Ol’šanskii, which says that there exist complementary series starting from Ind(ρ ⊗ ρ) whenever ρ is a unitary irreducible cuspidal representation. In appendix of [11], there is also a simple local proof of these results, based on a slightly different approach