On Intersections of Certain Partitions of a Group Compactification

Let G be a connected semi-simple algebraic group of adjoint type over an algebraically closed field, and let G 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 G, as well as intersections of B ×B-orbits and B− ×B−-orbits in G. We give explicit conditions for such intersections to be non-empty, and in each case, we show that every non-empty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the non-empty intersections form a strongly admissible partition of G.