Moment Lagrangians, unobstructedness and symplectic groupoids

Moment Lagrangian $L_{\mu }$ is a Lagrangian in $T^{\ast }{G}^{-}\times Y^{-}\times Y$ associated to a Hamiltonian $G$-space $Y$ with a moment map $\mu$. In this paper, we prove that $L_{\mu }$ is tautologically unobstructed under mild assumptions on $Y$. As a key ingredient in the proof, we constructed a new symplectic groupoid structure on $T^{\ast }{G}^{-}\times Y^{-}\times Y$ over $G\times Y$ for which $L_{\mu }$ is simultaneously the unit and the fixed locus of the inversion, which might be of independent interest.