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In this paper we study a natural generalization of symplectic toric manifoldsin the context of regular Poisson manifolds of compact types. To be moreprecise, we consider a class of multiplicity-free Hamiltonian actions byregular proper symplectic groupoids that we call faithful. Given such agroupoid, we classify its faithful multiplicity-free Hamiltonian actions interms of what we call Delzant subspaces of its orbit space -- certain`suborbifolds with corners' satisfying the Delzant condition relative to theintegral affine orbifold structure of the orbit space. This encompasses boththe classification of symplectic toric manifolds (due to Delzant) in terms ofDelzant polytopes and the classification of proper Lagrangian fibrations overan integral affine base manifold (due to Duistermaat) in terms of a sheafcohomology group. Each Delzant subspace comes with an orbifold version of thiscohomology, the degree one part of which classifies faithful multiplicity-freeHamiltonian actions with momentum map image equal to the Delzant subspace,provided there exists such an action. The obstruction to existence is encodedby a degree two class in this cohomology: the Lagrangian Dixmier-Douady class.In addition to the above, we introduce another invariant, which leads to avariation of our classification result involving only classical sheafcohomology and the group cohomology of certain modules for the isotropy groupsof the groupoid.

On the classification of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids