The moment map for a multiplicity free action

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.