Disintegrating tensor representations of nilpotent Lie groups
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- by Jawhar Abdennadher and Jean Ludwig PDF
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Abstract:
Let $G$ be a simply connected nilpotent Lie group and $H$ a closed connected subgroup of $G$. Given an irreducible unitary representation $\pi$ of $G$, we present an explicit disintegration of the restriction $\pi _{|H}$ of $\pi$ to $H$. Such a disintegration relies on the description of the double cosets space $H \diagdown G\diagup B$ for an arbitrary closed connected subgroup $B$ of $G$, and the well-known smooth disintegration of monomial representations of nilpotent Lie groups. As an application we get a concrete disintegration and a criterion of irreducibility for tensor products of a finite number of irreducible representations of $G$.References
- Didier Arnal, Hidenori Fujiwara, and Jean Ludwig, Opérateurs d’entrelacement pour les groupes de Lie exponentiels, Amer. J. Math. 118 (1996), no. 4, 839–878 (French, with French summary). MR 1400061, DOI 10.1353/ajm.1996.0032
- A. Baklouti and J. Ludwig, Désintégration des représentations monomiales des groupes de Lie nilpotents, J. Lie Theory 9 (1999), no. 1, 157–191 (French, with English summary). MR 1680003
- A. Baklouti and J. Ludwig, Entrelacement des restrictions des représentations unitaires des groupes de Lie nilpotents, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 2, 395–429 (French, with English summary). MR 1824959, DOI 10.5802/aif.1827
- P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard, and M. Vergne, Représentations des groupes de Lie résolubles, Monographies de la Société Mathématique de France, No. 4, Dunod, Paris, 1972. MR 0444836
- Lawrence Corwin and Frederick P. Greenleaf, Spectrum and multiplicities for restrictions of unitary representations in nilpotent Lie groups, Pacific J. Math. 135 (1988), no. 2, 233–267. MR 968611, DOI 10.2140/pjm.1988.135.233
- Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. MR 1070979
- Hidenori Fujiwara, Sur les restrictions des représentations unitaires des groupes de Lie résolubles exponentiels, Invent. Math. 104 (1991), no. 3, 647–654 (French). MR 1106754, DOI 10.1007/BF01245095
- Hidenori Fujiwara, Représentations monomiales des groupes de Lie nilpotents, Pacific J. Math. 127 (1987), no. 2, 329–352 (French). MR 881763, DOI 10.2140/pjm.1987.127.329
- A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
- George W. Mackey, The theory of unitary group representations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955. MR 0396826
- L. Pukánszky, Leçons sur les représentations des groupes, Monographies de la Société Mathématique de France, No. 2, Dunod, Paris, 1967 (French). MR 0217220
Additional Information
- Jawhar Abdennadher
- Affiliation: Département de Mathématiques, Faculté des Sciences de Sfax, RTE de Soukra KM 4. B. P. 802, 3018, Sfax, Tunisia
- Email: jawhar.abdennadher@fss.rnu.tn
- Jean Ludwig
- Affiliation: Département de Mathématiques, Laboratoire LMAM UMR 7122, Université de Metz, Ile du Saulcy, F-57045 Metz Cedex 1, France
- Email: ludwig@univ-metz.fr
- Received by editor(s): March 16, 2007
- Published electronically: September 29, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 819-848
- MSC (2000): Primary 22E27
- DOI: https://doi.org/10.1090/S0002-9947-08-04709-0
- MathSciNet review: 2452826