Picard groups in Poisson geometry

We study isomorphism classes of symplectic dual pairs $P\leftarrow S\rightarrow\overline{P}$, where $P$ is an integrable Poisson manifold, $S$ is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed $P$, these Morita self-equivalences of $P$ form a group ${\rm Pic}(P)$ under a natural “tensor product” operation. Variants of this construction are also studied, for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.