 
Summary: PARTITIONS OF THE WONDERFUL GROUP COMPACTIFICATION
JIANGHUA LU AND MILEN YAKIMOV
Abstract. We define and study a family of partitions of the wonderful compactification
G of a semisimple algebraic group G of adjoint type. The partitions are obtained from
subgroups of G × G associated to triples (A1, A2, a), where A1 and A2 are subgraphs of
the Dynkin graph of G and a: A1 A2 is an isomorphism. The partitions of G of
Springer and Lusztig correspond respectively to the triples (, , id) and (, , id).
1. Introduction
Let G be a connected semisimple algebraic group over an algebraically closed field k.
De Concini and Procesi [5, 6] constructed a wonderful compactification G of G, which is
a smooth irreducible (G × G)variety with finitely many (G × G)orbits. Let Gdiag be the
diagonal subgroup of G × G. In his study of parabolic character sheaves on G in [14, 15],
Lusztig introduced (by an inductive procedure) a partition of G by finitely many Gdiag
stable pieces. The closure of a Gdiagstable piece was shown by X.H. He [8] to be a union
of such pieces. Let B be a Borel subgroup of G. Then G is also partitioned into finitely
many (B × B)orbits. The (B × B)orbits in G, as well as their closures, were studied by T.
Springer in [18]. In [8], X.H. He gave a second description of Lusztig's Gdiagstable pieces
using (B × B)orbits in G, which then enabled him to give [9] an equivalent definition of
Lusztig's character sheaves on G. Further properties and applications of the Gdiagstable
pieces were obtained by X.H. He and J. F. Thomsen in [7, 9, 10].
