Optimal reduction

We generalize various symplectic reduction techniques of Marsden, Weinstein, Sjamaar, Bates, Lerman, Marle, Kazhdan, Kostant, and Sternberg to the context of the optimal momentum map. We see that, even though all those reduction procedures had been designed to deal with canonical actions on symplectic manifolds in the presence of a momentum map, our construction allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit reduction procedure for canonical actions on a Poisson manifold that exhibits an interesting interplay with the von Neumann condition previously introduced by the author in his study of the theory of singular dual pairs. More specifically, this condition ensures that the orbits in the momentum space of the optimal momentum map (we call them polar reduced spaces) admit a presymplectic structure that generalizes the Kostant–Kirillov–Souriau symplectic structure of the coadjoint orbits in the dual of a Lie algebra. Using this presymplectic structure, the optimal orbit reduced spaces are symplectic with a form that satisfies a relation identical to the classical one obtained by Marle, Kazhdan, Kostant, and Sternberg for free Hamiltonian actions on a symplectic manifold. In the Poisson case we provide a sufficient condition for the polar reduced spaces to be symplectic. In the symplectic case the polar reduced spaces are symplectic if and only if certain relation between the tangent space to the orbit and its symplectic orthogonal with the tangent space to the isotropy type submanifolds is satisfied. In general, the presymplectic polar reduced spaces are foliated by symplectic submanifolds that are obtained through a generalization to the optimal context of the so called Sjamaar Principle, already existing in the theory of Hamiltonian singular reduction. We call these subspaces the regularized polar reduced spaces. We use these ideas to shed some light in the problem of orbit reduction of globally Hamiltonian actions when the symmetry group is not compact and in the construction of a family of presymplectic homogeneous manifolds and of its symplectic foliation. We also show that these reduction techniques can be implemented in stages whenever we are in the presence of certain hypotheses that generalize those already existing for free globally Hamiltonian actions.