The G-stable pieces of the wonderful compactification

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Let G be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification 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 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.