Méthode des orbites pour les représentations de longueur finie. (The orbit method for representations of finite length) (Q912995)

The ``orbit method'' attaches unitary representations of a Lie group G to certain orbits of G on the dual \({\mathfrak g}^*\) of its Lie algebra. More precisely, in the formulation of \textit{M. Duflo} [Harmonic analysis and group representations, 129-221, Liguori/Naples (1982)], there is a functor T which, for \(f\in {\mathfrak g}^*\) ``admissible and well- polarizable'', sends a certain category X(G,f) of representations of a double cover \(G(f)^{\sim}\) of the stabilizer G(f) of f to unitary representations of G. Furthermore, T preserves irreducibility and distinct G-orbits in \({\mathfrak g}^*\) give disjoint representations of G. In this paper, the author shows that for G a motion group (i.e., a semidirect product of a vector group by a compact group), the functor T extends to the category of \(C^{\infty}\)-representations of finite length (where the composition factors have been obtained for G a connected nilpotent group by unitary representations). In the new context, most of the properties of the functor T carry over. However, it is necessary to restrict attention to orbits in general position. Similar results have been obtained for G a connected nilpotent group by \textit{F. du Cloux} [Astérisque 124-125, 129-211 (1985; Zbl 0567.22006)] and a more recent paper by du Cloux [Invent. Math. 88, 375-394 (1987; Zbl 0595.43008)].