Introduction

The construction of the reduced space for a symplectic manifold with symmetry, as formalized by Marsden and Weinstein, has proved to be very useful in many areas of mathematics ranging from classical mechanics to algebraic geometry. In the ideal situation, which requires the value of the moment map to be weakly regular, the reduced space is again a symplectic manifold. A lot of work has been done in the last ten years in the hope of finding a ‘correct’ reduction procedure in the case of singular values. For example, Arms, Gotay and Jennings describe several approaches to reduction in. At some point it has also been observed by workers in the field that in all examples the level set of a moment map modulo the appropriate group action is a union of symplectic manifolds. Recently Otto has proved that something similar does indeed hold, namely that such a quotient is a union of symplectic orbifolds. Independently two of us, R. Sjamaar and E. Lerman, have proved a stronger result. We proved that in the case of proper actions the reduced space, which we simply took to be the level set modulo the action, is a stratified symplectic space. Thereby we obtained a global description of the possible dynamics, a procedure for lifting the dynamics to the original space and a local characterization of the singularities of the reduced space. (The precise definitions will be given below.) The goal of this paper is twofold.