  We consider a connected compact Lie group K acting on a symplectic manifold M
such that a moment map m exists. A pull-back function via m Poisson commutes
with all K-invariants. Guillemin-Sternberg raised the problem to find a
converse. In this paper, we solve this problem by determining the Poisson
commutant of the algebra of K-invariants. It is completely controlled by the
image of m and a certain subquotient W_M of the Weyl group of K. The group W_M
is also a reflection group and forms a symplectic analogue of the little Weyl
group of a symmetric space. The proof rests ultimately on techniques from
algebraic geometry. In fact, a major part of the paper is of independent
interest: it establishes connectivity and reducedness properties of the fibers
of the (complex algebraic) moment map of a complex cotangent bundle.
