Poisson geometry of the Grothendieck resolution of a complex semisimple group

Abstract

We study a Poisson structure $\pi$ on the Grothendieck resolution $X$ of a complex semi-simple group $G$ and prove that the desingularization map $\mu :(X,\pi )\to (G,{\pi }_0)$ is Poisson, where ${\pi }_0$ is a Poisson structure such that intersections of conjugacy classes and opposite Bruhat cells $Bw{B}_{-}$ are Poisson subvarieties. We compute the symplectic leaves of $X$ and show that $(X,\pi )$ resolves singularities of $(G,{\pi }_0)$.

Final version for publication in MMJ, added references