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Journal of Lie Theory 29 (2019), No. 2, 375--390
Copyright Heldermann Verlag 2019

Trace Class Groups: the Case of Semi-Direct Products

Gerrit van Dijk
Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands


A Lie group $G$ is called a trace class group if for every irreducible unitary representation $\pi$ of $G$ and every $C^\infty$ function $f$ with compact support the operator $\pi (f)$ is of trace class. In this paper we extend the study of trace class groups, begun in a previous paper, to special families of semi-direct products. For the case of a semisimple Lie group $G$ acting on its Lie algebra $\mathfrak g$ by means of the adjoint representation we obtain a nice criterion in order that $\mathfrak g \rtimes G$ is a trace class group.

Keywords: Trace class group, Levi decomposition, semi-direct product, semisimple Lie group, orbit, invariant measure.

MSC: 22D10, 22E30, 43A80

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