Geodesic
Trace Class Groups: the Case of Semi-Direct Products
Journal of Lie theory, Tome 29 (2019) no. 2, pp. 375-39
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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.
Classification : 22D10, 22E30, 43A80
Mots-clés : Trace class group, Levi decomposition, semi-direct product, semisimple Lie group, orbit, invariant measure 
@article{JLT_2019_29_2_JLT_2019_29_2_a3,
     author = {G. van Dijk},
     title = {Trace {Class} {Groups:} the {Case} of {Semi-Direct} {Products}},
     journal = {Journal of Lie theory},
     pages = {375--39},
     year = {2019},
     volume = {29},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JLT_2019_29_2_JLT_2019_29_2_a3/}
}
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G. van Dijk. Trace Class Groups: the Case of Semi-Direct Products. Journal of Lie theory, Tome 29 (2019) no. 2, pp. 375-39. http://geodesic.mathdoc.fr/item/JLT_2019_29_2_JLT_2019_29_2_a3/