Poisson geometry of the Grothendieck resolution of a complex semisimple group

Let $G$ be a complex semi-simple Lie group with a fixed pair of opposite Borel subgroups $(B,B_{-})$. We study a Poisson structure $\pi$ on $G$ and a Poisson structure $\Pi$ on the Grothendieck resolution $X$ of $G$ such that the Grothendieck map $\mu\colon(X,\Pi)\to(G,\pi)$ is Poisson. We show that the orbits of symplectic leaves of $\pi$ in $G$ under the conjugation action by the Cartan subgroup $H=B\cap B$ – are intersections of conjugacy classes and Bruhat cells $B_{\omega}B_{-}$, while the $H$-orbits of symplectic leaves of $\Pi$ on $X$ give desingularizations of intersections of Steinberg fibers and Bruhat cells in $G$. We also give birational Poisson isomorphisms from quotients by $H\times H$ of products of double Bruhat cells in $G$ to intersections of Steinberg fibers and Bruhat cells.