Diagonal orbits in the wonderful compactification

TL;DR

This paper classifies diagonal orbits in the wonderful compactification of semisimple groups, studies their parameterization via maximal torus compactification, and constructs closures of Steinberg fibers, revealing their structure as unions of stable pieces.

Contribution

It provides a detailed classification of diagonal orbits in the wonderful compactification and links their structure to representations and stable pieces, advancing understanding of symmetric space compactifications.

Findings

Classification of certain diagonal orbits in the wonderful compactification.

Parameterization of these orbits via maximal torus compactification.

Construction and analysis of closures of Steinberg fibers as unions of stable pieces.

Abstract

The various types of compactifications of symmetric spaces and locally symmetric spaces are well-studied. Among them, the De Concini-Procesi compactification, also known as the wonderful compactification, of symmetric varieties has been found to have many applications. Intuitively, this compactification provides information at infinity. The diagonal action also extends the conjugation action on semisimple groups, which has received considerable attention. In this work, we will first describe the classification of certain diagonal orbits in the wonderful compactification of a semisimple adjoint group $ G $. We will then study the compactification of the maximal torus through representations of the simply connected cover $ \tilde{G} $, which, in a sense, parameterizes these diagonal orbits. Finally, we will focus on constructing the family of closures of the Steinberg fiber. We will examine the limit of this family and show that it is a union of He-Lusztig's $ G $-stable pieces.