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ArticleOriginal scientific text

Title

On the Moment Map of a Multiplicity Free Action

Authors 1, 2

Affiliations

  1. Department of Mathematics, Nicholas Copernicus University, 87-100 Toruń, Poland
  2. Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.

Abstract

The purpose of this note is to show that the Orbit Conjecture of C. Benson, J. Jenkins, R. L. Lipsman and G. Ratcliff [BJLR1] is true. Another proof of that fact has been given by those authors in [BJLR2]. Their proof is based on their earlier results, announced together with the conjecture in [BJLR1]. We follow another path: using a geometric quantization result of Guillemin-Sternberg [G-S] we reduce the conjecture to a similar statement for a projective space, which is a special case of a characterization of projective smooth spherical varieties due to Brion [B2].

Bibliography

  1. [BJLR1] C. Benson, J. Jenkins, R. L. Lipsman and G. Ratcliff, The moment map for a multiplicity-free action, Bull. Amer. Math. Soc. 31 (1994), 185-190.
  2. [BJLR2] C. Benson, J. Jenkins, R. L. Lipsman and G. Ratcliff, A geometric criterion for Gelfand pairs associated with the Heisenberg group, Pacific J. Math., to appear.
  3. [B1] M. Brion, Spherical Varieties: An Introduction, in: Topological Methods in Algebraic Transformation Groups, H. Kraft, T. Petrie and G. Schwarz (eds.), Progr. Math. 80, Birkhäuser, Boston, 1989, 11-26.
  4. [B2] M. Brion, Sur l'image de l'application moment, in: Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin, M.-P. Mallavin (ed.), Lecture Notes in Math. 1296, Springer, Berlin, 1987, 177-192.
  5. [G-S] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538.
  6. [O-V] A. L. Onishchik and E. B. Vinberg (eds.), Lie Groups and Lie Algebras III, Springer, Berlin, 1994.
  7. [Se] F. J. Servedio, Prehomogeneous vector spaces and varieties, Trans. Amer. Math. Soc. 176 (1973), 421-444.
Colloquium Mathematicum
1996/71/1
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Pages:
107-110
Main language of publication
English
Received
1995-10-30
Published
1996
Exact and natural sciences