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The Poisson Characteristic Variety of Unitary Irreducible Representations of Exponential Lie Groups

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 290)

Abstract

We recall the notion of Poisson characteristic variety of a unitary irreducible representation of an exponential solvable Lie group, and conjecture that it coincides with the Zariski closure of the associated coadjoint orbit. We prove this conjecture in some particular situations, including the nilpotent case.

Keywords

Poisson characteristic variety Representations Coadjoint orbits Exponential groups Solvable Lie algebras 

MSC Classification (2000)

22E27 81S10 

Notes

Acknowledgements

The authors would like to thank the Referee for having proposed some suggestions to improve the final form of the paper.

References

  1. 1.
    Baklouti, A., Dhieb, S., Manchon, D.: Orbites coadjointes et variétés caractéristiques. J. Geom. Phys. 54, 1–41 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baklouti, A., Dhieb, S., Manchon, D.: A deformation approach of the Kirillov map for exponential groups. Adv. Pure Appl. Math. 2, 421–436 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Corwin, L., Greenleaf, F.: Representations of nilpotent Lie groups and their applications. Part 1. Cambridge Studies in Advanced Mathematics, vol. 18. Cambridge University Press, Cambridge (2004)Google Scholar
  4. 4.
    Dixmier, J.: Algèbres Enveloppantes. Gauthier-Villars (1974)Google Scholar
  5. 5.
    Godfrey, C.: Ideals of coadjoint orbits of nilpotent Lie algebras. Trans. Am. Math. Soc. 233, 295–307 (1977)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Leptin, H., Ludwig, J.: Unitary Representations of Solvable Lie Groups. De Gruyter, Berlin (1994)Google Scholar
  7. 7.
    Mathieu, O.: Bicontinuity of the Dixmier map. J. Am. Math. Soc. 4(4), 837–863 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Pedersen, N.V.: On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. Bull. Soc. Math. France 112(42), 3–467 (1984)MathSciNetGoogle Scholar
  9. 9.
    Pedersen, N.V.: On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications. I. Math. Ann. 281, 633–699 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pedersen, N.V.: Orbits and primitive ideals of solvable Lie algebras. Math. Ann. 298(2), 275–326 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculté des Sciences de SfaxSfaxTunisia
  2. 2.LMBP, CNRSUniversité Clermont-AuvergneAubièreFrance

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