On intersections of certain partitions of a group compactification

Abstract

Let G be a connected semi-simple algebraic group of adjoint type over an algebraically closed field and let be the wonderful compactification of G. For a fixed pair (B,B-) of opposite Borel subgroups of G, we look at intersections of Lusztig's G-stable pieces and the B-×B-orbits in, as well as intersections of B×B-orbits and B-×B --orbits in. We give explicit conditions for such intersections to be nonempty, and in each case, we show that every nonempty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the nonempty intersections form a strongly admissible partition of G. © The Author(s) 2010. Published by Oxford University Press. All rights reserved.