Startseite On the generalized moment separability theorem for type 1 solvable Lie groups
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On the generalized moment separability theorem for type 1 solvable Lie groups

  • Lobna Abdelmoula , Ali Baklouti EMAIL logo und Yasmine Bouaziz
Veröffentlicht/Copyright: 17. Februar 2018
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Abstract

Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for π in G^, the unitary dual of G, sends smooth vectors of the representation space of π to 𝒰(𝔤)*, the dual vector space of 𝒰(𝔤). The convex hull of the image of the generalized moment map for π is called its generalized moment set, denoted by J(π). We say that G^ is generalized moment separable when the generalized moment sets differ for any pair of distinct irreducible unitary representations. Our main result in this paper provides a second proof of the generalized moment separability theorem for G.

MSC 2010: 22E27; 32G05

Funding statement: This work was completed with the support of D.G.R.S.R.T Research Laboratories: LR 11 ES 35 and LR 11 ES 52.

Acknowledgements

The authors are deeply grateful to the referee for useful comments and important suggestions which helped to improve the presentation of the paper.

References

[1] L. Abdelmoula, D. Arnal, J. Ludwig and M. Selmi, Separation of unitary representations of connected Lie groups by their moment sets, J. Funct. Anal. 228 (2005), no. 1, 189–206. 10.1016/j.jfa.2004.11.013Suche in Google Scholar

[2] D. Arnal, A. Baklouti, J. Ludwig and M. Selmi, Separation of unitary representations of exponential Lie groups, J. Lie Theory 10 (2000), 399–410. Suche in Google Scholar

[3] L. Auslander and B. Kostant, Quantization and representations of solvable Lie groups, Bull. Amer. Math. Soc. 73 (1967), 692–695. 10.1090/S0002-9904-1967-11829-9Suche in Google Scholar

[4] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255–354. 10.1007/BF01389744Suche in Google Scholar

[5] L. Baggett, Representations of the Mautner group. I, Pacific J. Math. 77 (1978), no. 1, 7–22. 10.2140/pjm.1978.77.7Suche in Google Scholar

[6] A. Baklouti, J. Ludwig and M. Selmi, Séparation des représentations unitaires des groupes de Lie nilpotents, Lie Theory and its Applications in Physics II (Clausthal 1997), World Scientific, Singapur (1996). Suche in Google Scholar

[7] P. Bernat, Sur les représentations unitaires des groups de Lie résolubles, Ann. Sci. Éc. Norm. Supér. (3) 82 (1965), 37–99. 10.24033/asens.1136Suche in Google Scholar

[8] P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard and M. Vergne, Représentations des groupes de Lie résolubles, Dunod, Paris, 1972. Suche in Google Scholar

[9] M. Duflo, Caractères des groupes et des algèbres de Lie résolubles, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 23–74. 10.24033/asens.1187Suche in Google Scholar

[10] H. Leptin and J. Ludwig, Unitary Representation Theory of Exponential Lie Groups, De Gruyter Exp. Math. 18, Walter de Gruyter, Berlin, 1994. 10.1515/9783110874235Suche in Google Scholar

Received: 2017-04-03
Revised: 2017-11-28
Accepted: 2017-12-04
Published Online: 2018-02-17
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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