Poisson Geometry of the Grothendieck Resolution of a Complex Semisimple Group

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) → (G,π₀) is Poisson, where π₀ 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, π₀).