Weyl Groups of Hamiltonian Manifolds, I

Let K be a compact connected Lie group and M a compact Hamiltonian K-manifold, i.e., a symplectic K-manifold equipped with a moment map μ : M → k. In this paper, we determine Col(M): the set of all functions on M which Poisson commute with all Kinvariant functions. For this, we construct a finite reflection group WM and show that Col(M) is completely determined by μ(M) and WM . More precisely, from μ(M) and WM we construct a topological space Y equipped with a differentiable structure (in fact, Y is semi-analytic) and a surjective map μ̂ : M → Y such that Col(M) consists exactly of the pull-back functions via μ̂. It is easy to see that, conversely, C(M) is the Poisson centralizer of Col(M). Thus we obtain a symplectic dual pair