zbMATH — the first resource for mathematics

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 \(\mathfrak g\).
22E25 Nilpotent and solvable Lie groups
Full Text: DOI
[1] Auslander, L.; Moore, C.C., Unitary representations of solvable Lie groups, Mem. amer. math. soc., 62, 199, (1966) · Zbl 0204.14202
[2] Bernat, P.; Conze, N.; Duflo, M.; Lévy-Nahas, M.; Raïs, M.; Renouard, P.; Vergne, M., Représentations des groupes de Lie résolubles, Monographies de la société mathématique de France, vol. 4, (1972), Dunod Paris · Zbl 0248.22012
[3] Duflo, M.; Raïs, M., Sur l’analyse harmonique sur LES groupes de Lie résolubles, Ann. sci. école norm. sup. (4), 9, 1, 107-144, (1976) · Zbl 0324.43011
[4] Fujiwara, H., Sur LES restrictions des représentations unitaires des groupes de Lie résolubles exponentiels, Invent. math., 104, 3, 647-654, (1991) · Zbl 0702.22012
[5] Mackey, G.W., Induced representations of locally compact groups. I, Ann. of math. (2), 55, 101-139, (1952) · Zbl 0046.11601
[6] Vargas, J., Restrictions of some unitary representations of \(\operatorname{SU}(n, 1)\) to \(\operatorname{U}(n - 1, 1)\), J. of funct. anal., 103, 2, 352-371, (1992) · Zbl 0787.22015
[7] Warner, G., Harmonic analysis on semi-simple Lie groups. I, Die grundlehren der mathematischen wissenschaften, Band 188, (1972), Springer-Verlag New York · Zbl 0265.22020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.