# On the Moment Map of a Multiplicity Free Action

Andrzej Daszkiewicz; Tomasz Przebinda

Colloquium Mathematicae (1996)

- Volume: 71, Issue: 1, page 107-110
- ISSN: 0010-1354

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topDaszkiewicz, Andrzej, and Przebinda, Tomasz. "On the Moment Map of a Multiplicity Free Action." Colloquium Mathematicae 71.1 (1996): 107-110. <http://eudml.org/doc/210415>.

@article{Daszkiewicz1996,

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].},

author = {Daszkiewicz, Andrzej, Przebinda, Tomasz},

journal = {Colloquium Mathematicae},

keywords = {spherical variety; connected reductive complex group; positive definite hermitian form; moment map; Lie algebra; orbit conjecture},

language = {eng},

number = {1},

pages = {107-110},

title = {On the Moment Map of a Multiplicity Free Action},

url = {http://eudml.org/doc/210415},

volume = {71},

year = {1996},

}

TY - JOUR

AU - Daszkiewicz, Andrzej

AU - Przebinda, Tomasz

TI - On the Moment Map of a Multiplicity Free Action

JO - Colloquium Mathematicae

PY - 1996

VL - 71

IS - 1

SP - 107

EP - 110

AB - 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].

LA - eng

KW - spherical variety; connected reductive complex group; positive definite hermitian form; moment map; Lie algebra; orbit conjecture

UR - http://eudml.org/doc/210415

ER -

## References

top- [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. Zbl0829.22014
- [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. Zbl0868.22015
- [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.
- [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.
- [G-S] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538. Zbl0503.58018
- [O-V] A. L. Onishchik and E. B. Vinberg (eds.), Lie Groups and Lie Algebras III, Springer, Berlin, 1994.
- [Se] F. J. Servedio, Prehomogeneous vector spaces and varieties, Trans. Amer. Math. Soc. 176 (1973), 421-444. Zbl0266.20043

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