Unipotent variety in the group compactification

The unipotent variety of a reductive algebraic group G plays an important role in the representation theory. In this paper, we will consider the closure of the unipotent variety in the De Concini–Procesi compactification of a connected simple algebraic group G. We will prove that is a union of some G-stable pieces introduced by Lusztig in [Moscow Math. J 4 (2004) 869–896]. This was first conjectured by Lusztig. We will also give an explicit description of . It turns out that similar results hold for the closure of any Steinberg fiber in .