On the centralizer of invariant functions on a Hamiltonian GG-space

A Hamiltonian action of a Lie group G on a symplectic manifold M gives rise to a moment map J: M -• g*, where 0* is the dual space to the Lie algebra g of G. The functions on M that are pullbacks of functions on g* by the moment map form a Poisson subalgebra of C^∞(M). Such functions are called collective. Assume that G and M are compact and connected. It is easy to see that the centralizer of collective functions in C^∞(M) consists of G-invariant functions. It was conjectured by Guillemin and Sternberg in [3] that the converse is also true, namely that the centralizer of the invariants is the set of collective functions. The main result of this paper is the proof of the conjecture in the case where the image of the moment map misses the walls of Weyl chambers in g*, i.e., when the stabilizers under the coadjoint action of the points in J(M) are all tori. An example shows that if J(M) intersects the walls, the conjecture may fail.