Singular dual pairs

AbstractWe generalize the notions of dual pair and polarity introduced by S. Lie (1890) and A. Weinstein (1983) in order to accommodate very relevant situations where the application of these ideas is desirable. The new notion of polarity is designed to deal with the loss of smoothness caused by the presence of singularities that are encountered in many problems in Poisson and symplectic geometry. We study in detail the relation between the newly introduced dual pairs, the quantum notion of Howe pair, and the symplectic leaf correspondence of Poisson manifolds in duality. The dual pairs arising in the context of symmetric Poisson manifolds are treated with special attention. We show that in this case and under very reasonable hypotheses we obtain a particularly well behaved kind of dual pairs that we call von Neumann pairs. Some of the ideas that we present in this paper shed some light on the optimal momentum maps introduced in [J.-P. Ortega, T.S. Ratiu, The optimal momentum map, in: P. Holmes, P. Newton, A. Weinstein (Eds.), Geometry, Dynamics and Mechanics: 60th Birthday Volume for J.E. Marsden, Springer-Verlag, New York, 2002, pp. 319–362]