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Duflo conjecture for solvable Lie groups. (La conjecture de Duflo pour les groupes résolubles exponentiels.) (French) Zbl 1195.22003
Summary: Let \(G\) be an exponential solvable Lie group, \(\mathfrak g\) its Lie algebra and \(\pi \) a unitary irreducible representation of \(G\) which is square integrable modulo the center, associated by the Kirillov-Bernat map [L. Auslander and C. C. Moore, Mem. Am. Math. Soc. 62, 199 p. (1966; Zbl 0204.14202); P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard and M. Vergne, Représentations des groupes de Lie résolubles, Paris: Dunod (1972; Zbl 0248.22012)] to a \(G\)-orbit \(\Omega \). Let \(H\) be a closed connected subgroup of \(G\) with Lie algebra \(\mathfrak h\) and \(p: \mathfrak g^* \to \mathfrak h^*\) the restriction map. We say that the representation \(\pi \) is \(H\)-admissible if its restriction to the subgroup \(H\) splits in irreducible representations with finite multiplicities. We prove the following conjecture due to Duflo: The representation \(\pi \) is \(H\)-admissible, if and only if, the restriction of \(p\) to \(\Omega \) is proper on the range \(p(\Omega )\). In the case at hand, these two conditions are equivalent to \(\mathfrak g= \mathfrak h + \mathfrak j\), where \(\mathfrak j\) is the center of \(\mathfrak g\).
MSC:
22E25 Nilpotent and solvable Lie groups
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