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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:
 2.2e+26 Nilpotent and solvable Lie groups
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##### References:
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