Double Bruhat cells and symplectic groupoids

Let $G$ be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure $pi_{{rm st}}$ determined by a pair of opposite Borel subgroups $(B, B_-)$. We prove that for each $v$ in the Weyl group $W$ of $G$, the double Bruhat cell $G^{v,v} = BvB cap B_-vB_-$ in $G$, together with the Poisson structure $pi_{{rm st}}$, is naturally a Poisson groupoid over the Bruhat cell $BvB/B$ in the flag variety $G/B$. Correspondingly, every symplectic leaf of $pi_{{rm st}}$ in $G^{v,v}$ is a symplectic groupoid over $BvB/B$. For $u, v in W$, we show that the double Bruhat cell $(G^{u,v}, pi_{{rm st}})$ has a naturally defined left Poisson action by the Poisson groupoid $(G^{u, u},pi_{{rm st}})$ and a right Poisson action by the Poisson groupoid $(G^{v,v}, pi_{{rm st}})$, and the two actions commute. Restricting to symplectic leaves of $pi_{{rm st}}$, one obtains commuting left and right Poisson actions on symplectic leaves in $G^{u,v}$ by symplectic leaves in $G^{u, u}$ and in $G^{v,v}$ as symplectic groupoids.