  Let $(M,\omega)$ be a connected symplectic manifold on which a connected Lie
group $G$ acts properly and in a Hamiltonian fashion with moment map $\mu:M
\lra \mf g^*$. Our purpose is investigate multiplicity-free actions, giving
criteria to decide a multiplicity freenes of the action. As an application we
give the complete classification of multiplicity-free actions of compact Lie
groups acting isometrically and in a Hamiltonian fashion on Hermitian symmetric
spaces of noncompact type. Successively we make a connection between
multiplicity-free actions on $M$ and multiplicity-free actions on the
symplectic reduction and on the symplectic cut, which allow us to give new
examples of multiplicity-free actions.
