  Let Phi : M --> g^* be a proper moment map associated to an action of a
compact connected Lie group, G, on a connected symplectic manifold, (M,\omega).
A collective function is a pullback via \Phi of a smooth function on g^*. In
this paper we present four new results about the relationship between the
collective functions and the G-invariant functions in the Poisson algebra of
smooth functions on M.
