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

Representation Theory