The partial compactification of the universal centralizer

The universal centralizer of a semisimple algebraic group G is the family of centralizers of regular elements, parametrized by their conjugacy classes. When G is of adjoint type, we construct a smooth, log-symplectic fiberwise compactification (Z) over bar of the universal centralizer (Z) over bar by taking the closure of each fiber in the wonderful compactification (G) over bar. We use the geometry of the wonderful compactification to give an explicit description of the symplectic leaves of (Z) over bar. We also show that its compactified centralizer fibers are isomorphic to certain Hessenberg varieties-we apply this connection to compute the singular cohomology of (Z) over bar, and to study the geometry of the corresponding universal Hessenberg family.