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Sur les caractères des groupes de Lie. (On the characters of Lie groups). (French) Zbl 0679.22009
Let G be a Lie group with Lie algebra \({\mathfrak g}\). Let \(\pi\) be a representation of G. The author studies the factor generated by \(\pi\) (G). The main result of the paper is the existence of operators of Hilbert-Schmidt class of the form \(\pi (u^ k*\phi)\) for \(\phi\in {\mathcal D}(G)\) and \(u\in \hat I\), where \(\hat I\) is the intersection of all the primitive ideals of U(\({\mathfrak g})\), containing the kernel I of the infinitesimal representation of U(\({\mathfrak g})\), corresponding to the representation \(\pi\). In the case of the normal factor representation \(\pi\) the author proves the existence of a trace class operator of the form \(\pi\) (\(\phi)\) for some \(\phi\in {\mathcal D}(G)\) relative to the factor generated by \(\pi\) (G).
Reviewer: S.Prishchepionok

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
17B15 Representations of Lie algebras and Lie superalgebras, analytic theory
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
17B35 Universal enveloping (super)algebras
46L35 Classifications of \(C^*\)-algebras
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