On unitary representations of algebraic groups over local fields

Bekka, Bachir; Echterhoff, Siegfried

doi:10.1090/ert/574

On unitary representations of algebraic groups over local fields
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by Bachir Bekka and Siegfried Echterhoff

Represent. Theory 25 (2021), 508-526

DOI: https://doi.org/10.1090/ert/574

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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