Diagonal orbits in the wonderful compactification
Yunsong Wei
TL;DR
The paper develops a framework to study diagonal $G$-stable orbits and Steinberg-fiber closures in the wonderful compactification $X$ of a semisimple adjoint group, connecting invariant-theoretic equations to boundary strata. It introduces a torus-compactification approach via representations of the simply connected cover $\tilde{G}$, showing the torus compactification depends only on the weight-support and establishing normality for key toric closures. A central construction is the family $Y\subset X\times \mathbb{C}^l$ defined by trace-based equations $f_{z,e_1,e_2,k}$, whose general fiber models the closure of a Steinberg fiber; the central fiber decomposes into $G$-stable pieces under explicit height and weight conditions. The work provides partial proofs of a central conjecture for types $G_2$, $B_l$, and $C_l$, describes the fibers at torus-vertices and the semistable locus, and outlines recursive toric-structure patterns for classical groups, offering a concrete geometric and representation-theoretic toolkit for understanding compactifications, orbit stratifications, and degenerations of conjugacy classes. The results have implications for parameterizing adjoint orbits, understanding boundary geometry, and linking Steinberg fibers with GIT quotients in the wonderful compactification setting.
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.