Multiplicity-free spaces

Let G be a Lie group. A unitary representation of G on a Hubert space, 77, is called multiplicty-free if every irreducible representation of G occurs in H with multiplicity zero or one. It is easy to see that this is the case if and only if the ring of bounded G-invariant operators on H is commutative. In this paper we will examine the symplectic analogue of this situation: Let X be a symplectic manifold on which G acts in a Hamiltonian fashion. If one thinks of the bounded operators on H as "quantum observables" and the functions on X as "classical observables" the analogue of the situation above is that the ring of G-invariant functions on X be commutative with respect to Poisson-bracket. If this happens we will say that X is multiplicity-free. We were led to the study of such manifolds by some questions in dynamical systems. Let : X - g* be the moment mapping. A function of the type / , for /: g* - R, is called collective (cf. 6); a completely integrable system consisting of functions of this type is called a collective completely integrable system (see We noticed We also proved that for certain groups, in particular for U(n) and O(n\ this condition is sufficient as well.