The Centralizer of Invariant Functions and Division Properties of the Moment Map

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.