The partial compactification of the universal centralizer

TL;DR

This paper constructs a smooth, log-symplectic partial compactification of the universal centralizer for adjoint type semisimple groups, describing its geometry and cohomology using the wonderful compactification and Hessenberg varieties.

Contribution

It introduces a novel partial compactification of the universal centralizer, linking it to Hessenberg varieties and providing explicit geometric and cohomological descriptions.

Findings

The compactified fibers are isomorphic to Hessenberg varieties.

Explicit description of symplectic leaves in the compactification.

Computed the singular cohomology of the partial compactification.

Abstract

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.