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Zbl 0760.22010
Charbonnel, Jean-Yves
Orbites fermées et orbites tempérées. (Closed orbits and tempered orbits). (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 23, No. 1, 123-149 (1990). ISSN 0012-9593

Let $G$ be a connected Lie group, ${\germ g}$ its real Lie algebra, and ${\germ g}\sp*$ the linear dual space of ${\germ g}$. The conjugation action of $G$ on itself leads to the adjoint representation of $G$ on ${\germ g}$ and then to the coadjoint representation of $G$ on ${\germ g}\sp*$. The resulting orbit structure of ${\germ g}\sp*$ is closely connected with the representation theory of $G$. In this connection, Pukanszky's work in the case of a nilpotent group $G$ shows that every coadjoint orbit of $G$ is tempered in the sense that integration over it, relative to the natural $G$-invariant measure, is a tempered distribution. For connected Lie groups in general, this is related to the question of whether the distribution character of a representation is tempered, and that is tightly tied to the question of whether the representation is weakly contained in the regular representation. So it is important to know whether a coadjoint orbit is tempered.\par For nilpotent $G$ the coadjoint orbits are automatically closed. The result that closed coadjoint orbits are tempered was extended to increasingly general classes of solvable Lie groups $G$ by a number of authors. There the final result, due to Pukanszky, is that if $G$ is solvable, and if the image of the adjoint representation of $G$ is a real linear algebraic group, then every closed coadjoint orbit of $G$ is tempered. If $G$ is a connected reductive (e.g. semisimple) Lie group, then the closed coadjoint orbits are just the semisimple coadjoint orbits, and they are the ones associated to the various series of representations constructed by Harish-Chandra. Harish-Chandra (and, independently, Trombi) proved that these are exactly the tempered representations of the reductive group $G$. It follows immediately that the closed coadjoint orbits are tempered. In this paper, the author succeeds in combining the results for Ad-algebraic and semisimple groups, proving in general for connected Lie groups that every closed coadjoint orbit is tempered.

MSC 2000:: *22E45 Analytic repres.of Lie and linear algebraic groups over real fields
22E46 Semi-simple Lie groups and their representations

Keywords: tempered orbits; Ad-algebraic groups; connected Lie group; coadjoint representation; semisimple groups; closed coadjoint orbit

Cited in: Zbl 0852.22013 Zbl 0856.22014 Zbl 0760.22011

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