Poisson geometry of the Grothendieck resolution of a complex semisimple group

Abstract

We study a Poisson structure $π$ on the Grothendieck resolution $X$ of a complex semi-simple group $G$ and prove that the desingularization map $μ:(X,π) \to (G,π_0)$ is Poisson, where $π_0$ is a Poisson structure such that intersections of conjugacy classes and opposite Bruhat cells $BwB_-$ are Poisson subvarieties. We compute the symplectic leaves of $X$ and show that $(X, π)$ resolves singularities of $(G, π_0)$.