Skip to main content
Log in

Invariant differential operators on symplectic grassmann manifolds

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

LetM 2n,r denote the vector space of real or complex2n×r matrices with the natural action of the symplectic group Sp 2n , and letG=G n,r =Sp 2n ×M 2n,r denote the corresponding semi-direct product. For any integerp with 0≤pn−1, letH denote the subgroupG p,r ×Sp 2n−2p ofG. We explicitly compute the algebra of left invariant differential operators onG/H, and we show that it is a free algebra if and only ifr2n−2p+1. We also give orthogonal analogues of these results, generalizing those of Gonzalez and Helgason [3].

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

Access this article

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

Bibliography

  1. Duflo, M.: Opérateurs invariants sur un espace symétrique. C. R. Acad. Sci. Paris289, 135–137 (1979)

    MATH  MathSciNet  Google Scholar 

  2. Helgason, S.: Differential operators on homogeneous spaces. Acta. Math.102, 239–299 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gonzalez, F., Helgason, S.: Invariant differential operators on Grassmann manifolds. Advances in Math.60, 81–91 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kraft, H.: Geometrische Methoden in der Invariantentheorie. Braunschweig: Vieweg-Verlag 1985.

    MATH  Google Scholar 

  5. Knop, F., Kraft, H., Vust, Th.: The Picard group of aG-variety. In: Algebraische Transformationsgruppen und Invariantentheorie (DMV-Seminar vol.13). Basel-Boston: Birkhäuser Verlag, 77–87 (1989)

    Google Scholar 

  6. Lichnèrowicz, A.: Opérateurs différentiels invariants sur un espace symétrique. C. R. Acad. Sci. Paris257, 3548–3550 (1963)

    Google Scholar 

  7. Luna, D., Vust, Th.: Un théorème sur les orbites affines des groupes algébriques semi-simples. Ann. Scuola Norm. Sup. Pisa27, 527–535 (1973)

    MathSciNet  Google Scholar 

  8. Popov, V.: Stability criteria for the action of a semi-simple group on a factorial manifold. Math. USSR Izvestija4, 527–535 (1970)

    Article  MATH  Google Scholar 

  9. Schwarz, G.: Representations of simple Lie groups with regular rings of invariants. Inv. Math.49, 167–191 (1978)

    Article  MATH  Google Scholar 

  10. Smoke, W.: Commutativity of the invariant differential operators on a symmetric space. Proc. Amer. Math. Soc.19, 222–224 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  11. Vust, Th.: Sur la théorie des invariants des groupes classiques. Ann. Inst. Fourier26, 1–31 (1976)

    MATH  MathSciNet  Google Scholar 

  12. Weyl, H.: The classical groups. Princeton: Princeton University Press 1946

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Partially supported by NSF grant DMS-9101358

This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schwarz, G., Zhu, Cb. Invariant differential operators on symplectic grassmann manifolds. Manuscripta Math 82, 191–206 (1994). https://doi.org/10.1007/BF02567697

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02567697

Keywords

Navigation