Sur les restrictions des représentations unitaires des groupes de Lie résolubles exponentiels. (On the restrictions of unitary representations of exponential solvable Lie groups). (French) Zbl 0702.22012

Let G be an exponential solvable Lie group, \(\pi\) an irreducible unitary representation of G, and let K be an analytic subgroup of G. We describe the canonical central decomposition of the restriction of \(\pi\) to K. This is done by the orbit method due to Kirillov. As corollary, we observe a Frobenius reciprocity and get the irreducible decomposition of the tensor product of unitary representations. These results were first shown by Corwin and Greenleaf in nilpotent case, then by Lipsman in completely solvable case.
Reviewer: H.Fujiwara


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


[1] Auslander, L., Moore, C.C.: Unitary representations of solvable Lie groups. Mem. Amer. Math. Soc. No. 62. 1966 · Zbl 0204.14202
[2] Bernat, P., et al.: Représentations des groupes de Lie résolubles. Paris: Dunod 1972
[3] Corwin, L., Greenleaf, F.P.: Spectrum and multiplicities for restrictions of unitary representations in nilpotent Lie groups. Pacific J. Math.135, 233-267 (1988) · Zbl 0628.22007
[4] Fujiwara, H.: Représentations monomiales des groupes de Lie résolubles exponentiels, 61-84 dans: Duflo, M., Pedersen, N.V., Vergne, M. (eds.) The orbit method in representation theory. Proceedings, Copenhagen 1988: Birkhäuser 1990
[5] Grélaud, G.: Désintégration des représentations induites des groupes de Lie résolubles exponentiels. Thèse de 3e cycle, Univ. de Poitiers 1973
[6] Kirillov, A.A.: Représentations unitaires des groupes de Lie nilpotents. Uspekhi Mat. Nauk17, 57-110 (1962)
[7] Lipsman, R.: Orbital parameters for induced and restricted representations. Trans. Amer. Math.313, 433-473 (1989) · Zbl 0683.22009
[8] Lipsman, R.: Induced representations of completely solvable Lie groups. à paraître dans Annali della Scuola Normale Superiore di Pisa · Zbl 0724.22006
[9] Lipsman, R.: Restricting representations of completely solvable Lie groups. à paraître dans Can. J. Math. · Zbl 0724.22007
[10] Mackey, G.W.: Induced representations of locally compact groups II: the Frobenius reciprocity theorem. Ann. Math.58, 193-221 (1953) · Zbl 0051.01901
[11] Mackey, G.W.: The theory of unitary group representations. Chicago Lect. Math., 1976 · Zbl 0344.22002
[12] Quint, S.R.: Decomposition of induced representations of solvable exponential Lie groups. Dissertation, Univ of California, Berkeley 1973
[13] Takenouchi, O.: Sur la facteur-représentation d’un groupe de Lie résoluble de type (E). Math. J. Okayama Univ.7, 151-161 (1957) · Zbl 0080.02302
[14] Vergne, M.: Étude de certaines représentations induites d’un groupe de Lie résoluble exponentiel. Ann. Sci. Ec. Norm. Sup.3, 353-384 (1970) · Zbl 0221.22014
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.