The Peterson Variety and the Wonderful Compactification

TL;DR

This paper establishes a geometric connection between the Peterson variety and the wonderful compactification by showing the closure of a centralizer of a regular nilpotent element is isomorphic to the Peterson variety, revealing new structural insights.

Contribution

It demonstrates that the closure of the centralizer of a regular nilpotent element in the wonderful compactification is isomorphic to the Peterson variety, providing a new geometric perspective.

Findings

The closure of the centralizer of a regular nilpotent element is isomorphic to the Peterson variety.

The closure of the centralizer of any regular element is isomorphic to the closure of a general orbit in the flag variety.

The paper describes the orbit structure of the Peterson variety under the action of the centralizer.

Abstract

We look at the centralizer in a semisimple algebraic group $G$ of a regular nilpotent element, and show that its closure in the wonderful compactification is isomorphic to the Peterson variety. It follows that the closure in the wonderful compactification of the centralizer $G^x$ of any regular element $x$ is isomorphic to the closure of a general $G^x$-orbit in the flag variety. We also give a description of the $G^e$-orbit structure of the Peterson variety.