Summary: PARTITIONS OF THE WONDERFUL GROUP COMPACTIFICATION
JIANG-HUA LU AND MILEN YAKIMOV
Abstract. We define and study a family of partitions of the wonderful compactification
G of a semi-simple 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).
Let G be a connected semi-simple 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 Gdiag-stable piece was shown by X.-H. He  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 . In , X.-H. He gave a second description of Lusztig's Gdiag-stable pieces
using (B × B)-orbits in G, which then enabled him to give  an equivalent definition of
Lusztig's character sheaves on G. Further properties and applications of the Gdiag-stable
pieces were obtained by X.-H. He and J. F. Thomsen in [7, 9, 10].