On the Set of Orbits for a Borel Subgroup

Let X = G/H be a homogeneous variety for a connected complex reductive group G and let B be a Borel subgroup of G. In many situations, it is necessary to study the B-orbits in X. An equivalent setting of this problem is to analyze H-orbits in the flag variety G/B. The probably best known example is the Bruhat decomposition of G/B where one takes H = B. Another well-studied situation is the case where H is a symmetric subgroup, i.e., the fixed point group of an involution of G. Then H-orbits in G/B play a very important role in representation theory. They are the main ingredients for the classification of irreducible Harish-Chandra modules (see e.g. the surveys [Sch], [Wo]). In this paper, we introduce two structures on the set of all B-orbits. The first one is not really new, namely an action of a monoid W ∗ on the set B(X) of all B-stable closed subvarieties of X. As a set, W ∗ is the Weyl group W of G but with a different multiplication. That has already been done by Richardson and Springer [RS1] in the case of symmetric varieties and the construction generalizes easily. As an application we obtain a short proof of a theorem of Brion [Br1] and Vinberg [Vin]: If B has a open orbit in X then B has only finitely many orbits. Varieties with this property are called spherical . All examples mentioned above are of this type. The second structure which we are introducing is an action of the Weyl group W on a certain subset of B(X). Let me remark that in the most important case, X spherical, B(X) is just the set of B-orbit closures and the W -action will be defined on all of it. We give two methods to construct this action. In the first, we define directly the action of the simple reflections sα of W . This is done by reduction to the case rkG = 1 and then by a case-by-case consideration. The advantage of this method is that it is very concrete and works in general. The problem is to show that the sα-actions actually define