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 over(U, -) of the unipotent variety in the De Concini-Procesi compactification over(G, -) of a connected simple algebraic group G. We will prove that over(U, -) - U 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 over(U, -). It turns out that similar results hold for the closure of any Steinberg fiber in over(G, -). © 2005 Elsevier Inc. All rights reserved