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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



The moment map for a multiplicity free action

Authors: Chal Benson, Joe Jenkins, Ronald L. Lipsman and Gail Ratcliff
Journal: Bull. Amer. Math. Soc. 31 (1994), 185-190
MSC: Primary 22C05; Secondary 22E30
MathSciNet review: 1260517
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Abstract: Let K be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space V. One calls this a multiplicity-free action whenever the K-isotypic components of $ \mathbb{C}{\text{[}}V]$ are K-irreducible. We have shown that this is the case if and only if the moment map $ \tau :V \to {\mathfrak{k}^{\ast} }$ for the action is finite-to-one on K-orbits. This is equivalent to a result concerning Gelfand pairs associated with Heisenberg groups that is motivated by the Orbit Method. Further details of this work will be published elsewhere.

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Keywords: Gelfand pairs, Heisenberg group, Orbit Method, moment map
Article copyright: © Copyright 1994 American Mathematical Society