Poisson geometry of the Grothendieck resolution of a complex semisimple group

Abstract

Let G be a complex semi-simple Lie group with a fixed pair of opposite Borel subgroups (B,B − ) . We study a Poisson structure π on G and a Poisson structure Π on the Grothendieck resolution X of G such that the Grothendieck map μ:(X,Π)→(G,π) is Poisson. We show that the orbits of symplectic leaves of π in G under the conjugation action by the Cartan subgroup H=B∩B – are intersections of conjugacy classes and Bruhat cells B ω B − , while the H -orbits of symplectic leaves of Π on X give desingularizations of intersections of Steinberg fibers and Bruhat cells in G . We also give birational Poisson isomorphisms from quotients by H×H of products of double Bruhat cells in G to intersections of Steinberg fibers and Bruhat cells.