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].
@article{bwmeta1.element.bwnjournal-article-cmv71i1p107bwm,
author = {Andrzej Daszkiewicz and Tomasz Przebinda},
title = {On the Moment Map of a Multiplicity Free Action},
journal = {Colloquium Mathematicae},
volume = {70},
year = {1996},
pages = {107-110},
zbl = {0854.22013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv71i1p107bwm}
}
Daszkiewicz, Andrzej; Przebinda, Tomasz. On the Moment Map of a Multiplicity Free Action. Colloquium Mathematicae, Tome 70 (1996) pp. 107-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv71i1p107bwm/
[000] [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. | Zbl 0829.22014
[001] [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. | Zbl 0868.22015
[002] [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.
[003] [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.
[004] [G-S] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538. | Zbl 0503.58018
[005] [O-V] A. L. Onishchik and E. B. Vinberg (eds.), Lie Groups and Lie Algebras III, Springer, Berlin, 1994.
[006] [Se] F. J. Servedio, Prehomogeneous vector spaces and varieties, Trans. Amer. Math. Soc. 176 (1973), 421-444. | Zbl 0266.20043