
Journal of Lie Theory 29 (2019), No. 2, 375390 Copyright Heldermann Verlag 2019 Trace Class Groups: the Case of SemiDirect Products Gerrit van Dijk Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands dijk@math.leidenuniv.nl [Abstractpdf] 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 semidirect 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, semidirect product, semisimple Lie group, orbit, invariant measure. MSC: 22D10, 22E30, 43A80 [ Fulltextpdf (153 KB)] for subscribers only. 