The Peterson variety and the wonderful compactification

Abstract

We look at the centralizer in a semisimple algebraic group G G of a regular nilpotent element e ∈ Lie ( G ) e\in \text {Lie}(G) 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 G^x of any regular element x ∈ Lie ( G ) x\in \text {Lie}(G) is isomorphic to the closure of a general G x G^x -orbit in the flag variety. We also give a description of the G e G^e -orbit structure of the Peterson variety.