  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.
