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In this paper we give a simple (local) proof of two principal results aboutirreducible tempered representations of general linear groups over anon-archimedean local division algebra. We give a proof of the parameterizationof the irreducible square integrable representations of these groups bysegments of cuspidal representations, and a proof of the irreducibility of thetempered parabolic induction. Our proofs are based on Jacquet modules (and theGeometric Lemma, incorporated in the structure of a Hopf algebra). We use onlysome very basic general facts of the representation theory of reductive p-adicgroups (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 thatthere exist complementary series starting from $Ind(\rho\otimes\rho)$ whenever$\rho$ is a unitary irreducible cuspidal representation. In appendix of thepaper "On parabolic induction on inner forms of the general linear group over anon-archimedean local field" of E. Lapid and A. Minguez, there is also a simplelocal proof of these results, based on a slightly different approach.

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