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The $G$-stable pieces of the wonderful compactification

Xuhua He
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Abstract

Let $G$ be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification $\bar{G}$ of $G$ into finite many $G$-stable pieces, which were introduced by Lusztig. In this paper, we will investigate the closure of any $G$-stable piece in $\bar{G}$. We will show that the closure is a disjoint union of some $G$-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many $G$-orbits.

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Xuhua He. 2006-02-01. The $G$-stable pieces of the wonderful compactification. https://arxiv.org/abs/math/0412302

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