Moment Lagrangians, unobstructedness and symplectic groupoids

RESUMO

Moment Lagrangian $L_\mu$ is a Lagrangian in $T^*G^- \times Y^- \times Y$ 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 $T^*G^- \times Y^- \times Y$ 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.