Momentum maps and Morita equivalence

We introduce quasi-symplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including ordinary Hamiltonian G-spaces, Lu's momentum maps of Poisson group actions, and the group-valued momentum maps of Alekseev-Malkin-Meinrenken. More precisely, we carry out the following program: (1) we define and study properties of quasi-symplectic groupoids. (2) We study the momentum map theory defined by a quasi-symplectic groupoid Gamma paired right arrows P. In particular, we study the reduction theory and prove that J(-1)(O)/Gamma is a symplectic manifold for any Hamiltonian F-space (X ->(J) P,omega(X)) (even though omega(X) is an element of Omega(2)(X) may be degenerate), where O subset of P is a groupoid orbit. More generally, we prove that the intertwiner space (X-1 x P <(X-2)overbar >)/Gamma between two Hamiltonian Gamma-spaces X-1 and X-2 is a symplectic manifold (whenever it is a smooth manifold). (3) we study Morita equivalence of quasi-symplectic groupoids. In particular, we prove that Morita equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories. Moreover the intertwiner space (X-1 x P <(X-2)over bar >)/Gamma depends only on the Morita equivalence class. As a result, we recover various well-known results concerning equivalence of momentum maps including the Alekseev-Ginzburg-Weinstein linearization theorem and the Alekseev-Malkin-Meinrenken equivalence theorem between quasi-Hamiltonian spaces and Hamiltonian loop group spaces.