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Double Bruhat cells and symplectic groupoids

Let $G$ be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure $π_{\rm st}$ 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 $π_{\rm st}$, is naturally a Poisson groupoid over the Bruhat cell $BvB/B$ in the flag variety $G/B$. Correspondingly, every symplectic leaf of $π_{\rm st}$ in $G^{v,v}$ is a symplectic groupoid over $BvB/B$. For $u, v \in W$, we show that the double Bruhat cell $(G^{u,v}, π_{\rm st})$ has a naturally defined left Poisson action by the Poisson groupoid $(G^{u, u},π_{\rm st})$ and a right Poisson action by the Poisson groupoid $(G^{v,v}, π_{\rm st})$, and the two actions commute. Restricting to symplectic leaves of $π_{\rm st}$, 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.