On the Moment Map of a Multiplicity Free Action

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].