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Representation Theory

ISSN 1088-4165



On unitary representations of algebraic groups over local fields

Authors: Bachir Bekka and Siegfried Echterhoff
Journal: Represent. Theory 25 (2021), 508-526
MSC (2020): Primary 22D10, 22D25, 22E50, 20G05
Published electronically: June 10, 2021
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Abstract: Let $\mathbf {G}$ be an algebraic group over a local field $\mathbf {k}$ of characteristic zero. We show that the locally compact group $\mathbf {G}(\mathbf {k})$ consisting of the $\mathbf {k}$-rational points of $\mathbf {G}$ is of type I. Moreover, we complete Lipsman’s characterization of the groups $\mathbf {G}$ for which every irreducible unitary representation of $\mathbf {G}(\mathbf {k})$ is a CCR representation and show at the same time that such groups belong to the family of trace class groups, recently studied by Deitmar and van Dijk.

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Additional Information

Bachir Bekka
Affiliation: Université Rennes, CNRS, IRMAR–UMR 6625, Campus Beaulieu, F-35042 Rennes Cedex, France
MR Author ID: 33840

Siegfried Echterhoff
Affiliation: Mathematisches Institut,Universität Münster, Einsteinstraße 62,, D-48149 Münster, Germany
MR Author ID: 266728
ORCID: 0000-0001-9443-6451

Received by editor(s): March 23, 2020
Published electronically: June 10, 2021
Additional Notes: The first author acknowledges the support by the ANR (French Agence Nationale de la Recherche) through the projects Labex Lebesgue (ANR-11-LABX-0020-01) and GAMME (ANR-14-CE25-0004). This research was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure
Article copyright: © Copyright 2021 by the authors