Order and Domains of Attraction of Control Sets in Flag Manifolds

Let G be a real semi-simple noncompact Lie group and S G a subsemigroup with int S = .This article relates the Bruhat-Chevalley order in the Weyl group W of G to the ordering of the control sets for S in the flag manifolds of G by showing that the one-to-one correspondence between the control sets and the elements of a double coset W (S) /W of W reverses the orders.This fact is used to show that the domain of attraction of a control set is a union of Schubert cells.