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On the mixed representations of exponential solvable Lie groups. (Sur les représentations mixtes des groupes de Lie résolubles exponentiels.) (French) Zbl 1015.22004

Let \(G\) be an exponential solvable Lie group, \(H\) an analytic subgroup of \(G\), and let \(\sigma\) be an irreducible unitary representation of \(H\). The authors consider the induced representation \(\tau=\text{Ind}^G_H \sigma\) of \(G\), and then restrict \(\tau\) to another analytic subgroup \(A\) of \(G\). The paper studies the canonical central decomposition of this restriction \(\tau|_A\). The authors determine in terms of the orbit method the measure class and the multiplicities for this irreducible decomposition [cf. the reviewer’s two articles: Prog. Math. 82, 61-84 (1990; Zbl 0744.22010); Invent. Math. 104, 647-654 (1991; Zbl 0716.22004)]. They proceed to examine in detail the case where \(G\) is nilpotent in order to show that the multiplicities in the irreducible decomposition of \(\tau|_A\) are either uniformly infinite or finite and bounded, and to get a necessary and sufficient condition for the latter eventuality [cf. R. L. Lipsman, Ann. Mat. Pura Appl. (4) 166, 291-300 (1994; Zbl 0829.43008)]. The same results hold for exponential \(G\) if \(H\) and \(A\) are normal subgroups in \(G\).

MSC:

22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
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