Rad HAZU, Matematičke znanosti, Vol. 19 (2015), 27-53.

REMARK ON REPRESENTATION THEORY OF GENERAL LINEAR GROUPS OVER A NON-ARCHIMEDEAN LOCAL DIVISION ALGEBRA

Marko Tadić

Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
e-mail: tadic@math.hr


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. Olíšanskii, 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.

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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