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

Let Φ : M −→ g be a proper moment map associated to an action of a compact connected Lie group, G, on a connected symplectic manifold, (M, ω). A collective function is a pullback via Φ 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 . More specifically, we show: 1. The centralizer of the invariant functions consists of the algebra of smooth functions on M that are constant on the level sets of the moment map. This resolves a conjecture of Guillemin and Sternberg. 2. The question of whether this centralizer is equal to the algebra of collective functions or is larger is equivalent to a formal algebraic question on the level of power series. 3. If the group G is a torus, the centralizer of the invariant functions consists of the collective functions. We close a gap in earlier proofs of this fact. 4. If the group G is SU(2) and the centralizer of the invariant functions is larger than the algebra of of collective functions, the action of SU(2) extends to an action of U(2) with the same orbits, and the centralizer of the invariant functions consists of the U(2)-collective functions.