The universal centralizer of a complex algebraic group G is the family of centralizers of regular elements in Lie(G), parametrized by their conjugacy classes. It has a natural symplectic structure, because it is obtained through a Whittaker Hamiltonian reduction from the cotangent bundle T*G. I will construct a smooth, log-symplectic partial compactification of the universal centralizer by taking the closure of each centralizer fiber in the wonderful compactification of G.

I will then show that these compactificatied centralizers can be identified with a class of subvarieties of the flag variety called Hessenberg varieties. This identification gives a natural log-symplectic Poisson structure on the universal family of Hessenberg varieties, which is a partial compactification of the twisted cotangent bundle of the base affine space. If there is time, I will describe approaches to a multiplicative version of these results.