The moment map for a multiplicity free action
  Let $K$ be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space $V$. One calls this a {\em multiplicity-free} action whenever the $K$-isotypic components of $\C[V]$ are $K$-irreducible. We have shown that this is the case if and only if the moment map $\tau:V\rightarrow\k^*$ for the action is finite-to-one on $K$-orbits. This is equivalent to a result concerning \gp s associated with Heisenberg groups that is motivated by the Orbit Method. Further details of this work will be published elsewhere. 
Chal Benson, Joe Jenkins, Ronald Lipsman, Gail Ratcliff