Skip to main content
SpringerLink
Log in

A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

Let G be a connected, simply connected real nilpotent Lie group with Lie algebra đ”€, H a connected closed subgroup of G with Lie algebra đ”„ and f a linear form on đ”€ satisfying f([đ”„, đ”„]) = {0}⋅ Let χ f be the unitary character of H with differential \({{\sqrt{{-1}}f}}\) at the origin. Let τ f be the unitary representation of G induced from the character χ f of H. We consider the algebra 𝒟(đ”€, đ”„, f) of differential operators invariant under the action of G on the bundle with basis G/H associated to these data. We show that 𝒟(đ”€, đ”„, f) is commutative if and only if τ f is of finite multiplicities. This proves a conjecture of Corwin-Greenleaf and Duflo.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baklouti, A., Fujiwara, H.: OpĂ©rateurs diffĂ©rentiels associĂ©s Ă  certaines reprĂ©sentations unitaires d’un groupe de Lie rĂ©soluble exponentiel. To appear in Compositio Math.

  2. Baklouti, A., Ludwig, J.: Invariant differential operators on certain nilpotent homogeneous spaces. Monatsh. Math. 134, 19–37 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Corwin, L., Greenleaf, F. P., GrĂ©laud, G.: Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups. Trans. Am. Math. Soc. 304, 549–583 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Corwin, L., Greenleaf, F. P.: A canonical approach to multiplicity formulas for induced and restricted representations of nilpotent Lie groups. Commun. Pure Appl. Math. 41, 1051–1088 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Corwin, L., Greenleaf, F. P.: Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity. Commun. Pure Appl. Math. 45, 681–748 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Duflo, M.: Open Problems in Representation Theory of Lie groups. Conference on ‘‘Analysis on homogeneous spaces’’, Katata, Japan, 1986, pp. 1–5

  7. Fujiwara, H., Lion, G., Magneron, B.: Sur certaines sous-algĂšbres de fonctions associĂ©es aux reprĂ©sentations monomiales des groupes de Lie nilpotents. PrĂ©publications de l’UniversitĂ© Paris-Nord, 2002-02, 2002

  8. Fujiwara, H., Lion, G., Mehdi, S.: On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces. Trans. Am. Math. Soc. 353, 4203–4217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fujiwara, H., Lion, G., Magneron, B., Mehdi, S.: Un critĂšre de commutativitĂ© pour l’algĂšbre des opĂ©rateurs diffĂ©rentiels invariants sur un espace homogĂšne nilpotent. C. R. Acad. Sci. Paris SĂ©r. I Math. 332, 597–600 (2001)

    Article  MATH  Google Scholar 

  10. Lipsman, R.: Orbital parameters for induced and restricted representations. Trans. Am. Math. Soc. 313, 433–473 (1989)

    MathSciNet  MATH  Google Scholar 

  11. Pukanszky, L.: Leçons sur les représentations des groupes. Monographies de la Société Mathématique de France. No 2. Dunod, Paris, 1967

Download references

Author information

Authors and Affiliations

  1. Department of industrial Engineering and Management, Kyushu School of Engineering, Kinki University, Iizuka, 820-8555, Japan

    H. Fujiwara

  2. Équipe Modal’X, UniversitĂ© Paris X, 200 Avenue de la RĂ©publique, 92001, Nanterre, France

    G. Lion

  3. Institut Galilée, Département de Mathématiques, Université Paris XIII, Avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    B. Magneron

  4. Mathematisches Institut, Georg-August UniversitÀt, Bunsenstrasse 3-5, 37073, Göttingen, Germany

    S. Mehdi

  5. Équipe de ThĂ©orie des Groupes, Institut de MathĂ©matiques de Jussieu, UniversitĂ© Paris VII, 2 Place Jussieu, 75251, Paris Cedex 05, France

    G. Lion, B. Magneron & S. Mehdi

Authors
  1. H. Fujiwara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Lion
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Magneron
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Mehdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Magneron.

Additional information

Mathematics Subject Classification (1991): 43A80, 43A85, 22E25, 22E27, 22E30

UMR nˆ 7539 du CNRS, ‘‘Analyse, GĂ©omĂ©trie et Applications’’.

UMR nˆ 7586 du CNRS, ‘‘ThĂ©orie des Groupes, ReprĂ©sentations, Applications’’.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fujiwara, H., Lion, G., Magneron, B. et al. A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces. Math. Ann. 327, 513–544 (2003). https://doi.org/10.1007/s00208-003-0464-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-003-0464-3

Keywords

Search

Navigation