Source: https://web.math.pmf.unizg.hr/~duje/radhazumz/vol19/no19_03.html
Timestamp: 2019-04-22 16:28:00+00:00

Document:
Rad HAZU, Matematičke znanosti, Vol. 19 (2015), 27-53.
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. 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 , there is also a simple local proof of these results, based on a slightly different approach.
2010 Mathematics Subject Classification. 22E50.
Key words and phrases. Non-archimedean local fields, division algebras, general linear groups, Speh representations, parabolically induced representations, reducibility, unitarizability.
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