Advertisement

Monomial Representations of Discrete Type of an Exponential Solvable Lie Group

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

Abstract

Let G be an exponential solvable Lie group, H an analytic subgroup of G and \(\chi \) a unitary character of H. We study some problems related to the induced representation \(\tau = \text {ind}_H^G \chi \) of G when \(\tau \) has multiplicities either finite or infinite of discrete type. In particular, we are interested in the Plancherel formula for \(\tau \) and the commutativity problem due to Duflo (Open problems in representation theory of Lie groups, edited by T. Oshima, Katata, Japan 1986, [12]) for the algebra \(D_{\tau }(G/H)\) of G-invariant differential operators on the fiber space associated to the data \((H,\chi )\) over the base space G / H. We give in particular an example where this problem can admit a negative solution in the frame of exponential solvable Lie groups.

Keywords

Orbit method Irreducible representations Penney distribution Plancherel formula Differential operator 

1991 Mathematics Subject Classication

22E27 

Notes

Acknowledgements

The authors would like to thank the Referee for having proposed many valuable comments and suggestions to improve the final form of the paper.

References

  1. 1.
    Arnal, D., Fujiwara et, H., Ludwig, J.: Opérateurs d’entrelacement pour les groupes de Lie exponentiels, Amer. J. Math. 118, 839–878 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baklouti, A., Ludwig, J.: Désintégration des représentations monomiales des groupes de Lie nilpotents. J. Lie Theory 9, 157–191 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baklouti et, A., Fujiwara, H.: Opérateurs différentiels associés à certaines représentations unitaires des groupes de Lie résolubles exponentiels. Compositio. Math. 139, 29–65 (2003)Google Scholar
  4. 4.
    Baklouti, A., Hamrouni, H., Khlif, F.: Analysis of some monomial representations of exponential solvable Lie groups. Russ. J. Math. Phys. 13(4), 363–379 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baklouti, A., Hamrouni, H.: The multiplicity function of mixed representations on completely solvable lie groups. Tokyo. J. Math. 30(1), 41–55 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bernat, P., et al.: Représentations des groupes de Lie résolubles. Dunod, Paris (1972)zbMATHGoogle Scholar
  7. 7.
    Cartier, P.: Vecteurs différentiables dans les représentations unitaires des groupes de Lie. Lecture Notes in Mathematics, vol. 514, pp. 20–34. Springer, Berlin (1975)Google Scholar
  8. 8.
    Corwin, L., Greenleaf, F.P.: Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity. Comm. Pure Appl. Math. 45, 681–748 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dixmier, J.: Algèbres enveloppantes. Gauthier-Villars, Paris (1974)zbMATHGoogle Scholar
  10. 10.
    Dixmier, J., Malliavin, P.: Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull. Sci. Math. 102, 305–330 (1978)zbMATHGoogle Scholar
  11. 11.
    Duflo, M.: Personal CommunicationGoogle Scholar
  12. 12.
    Duflo, M.: Open problems in representation theory of Lie groups, edited by T. Oshima. pp. 1–5. Katata, Japan (1986)Google Scholar
  13. 13.
    Fujiwara, H.: Certains opérateurs d’entrelacement pour des groupes de Lie résolubles exponentiels et leurs applications. Mem. Fac. Sci. Kyushu Univ. Ser A 36, 13–72 (1982)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fujiwara, H.: Représentations monomiales des groupes de Lie nilpotents. Pacific J. Math. 127, 329–351 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fujiwara, H.: Représentations monomiales des groupes de Lie résolubles exponentiels, The orbit method in representation theory. In: Duflo, M., Pedersen, N.V., Vergne, M. (eds.) Proceedings of a conference in Copenhagen, pp. 61–84. Birkhaüser, Boston (1990)CrossRefGoogle Scholar
  16. 16.
    Fujiwara et, H., Yamagami, S.: Certaines représentations monomiales d’un groupe de Lie résoluble exponentiel. Adv. St. Pure Math. 14, 153–190 (1988)Google Scholar
  17. 17.
    Fujiwara, H., Lion, G., Magneron, B., Mehdi, S.: Commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces. Math. Ann. 327, 513–544 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Grélaud, G.: Désintégration des représentations induites des groupes de Lie résolubles exponentiels, Thèse de \(3^e\) cycle, Univ. de Poitiers (1973)Google Scholar
  19. 19.
    Lipsman, R.: The Penney-Fujiwara Plancherel formula for homogeneous spaces, in representation theory of lie groups and lie algebras. In: Proceedings of Fuji-Kawaguchiko Conference, pp. 120–139. World-Scientific Publishing Co., Singapore (1992)Google Scholar
  20. 20.
    Leptin, H., Ludwig, J.: Unitary Representation Theory of Exponential Lie Groups. W. De Gruyter, Berlin (1994)CrossRefGoogle Scholar
  21. 21.
    Penney, R.: Abstract Plancherel theorem and a Frobenius reciprocity theorem. J. Funct. Anal. 18, 177–190 (1975)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Poulsen, N.S.: On \(C^{\infty }\)-vectors and intertwining bilinear forms for representations of lie groups. J. Funct. Anal. 9, 87–120 (1972)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Vergne, M.: Étude de certaines représentations induites d’un groupe de Lie résoluble exponentiel. Ann. Sci. Éc. Norm. Sup. 3, 353–384 (1970)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Département de MathématiquesFaculté des Sciences de SfaxSfaxTunisie
  2. 2.Faculty of Science and Technology for HumanityKinki UniversityIizukaJapan
  3. 3.Laboratoire LMAM, UMR 7122, Université de MetzMetz Cedex 01France

Personalised recommendations