  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.
