On the Set of Orbits for a Borel Subgroup

Let G be a connected reductive group with Borel subgroup B and let X be a G-variety. We study the set of B-orbits in X. In case X=G/H is homogeneous, this is the same as to study H-orbits in the flag variety G/B. Assume for simplicity that B has an open orbit in X. First, we give a new proof that then X contains in fact only finitely many B-orbits. Let W be the Weyl group of G. Then, the main part is to introduce and study a W-action on this finite set of B-orbits. The W-isotropy group of the open B-orbit is linked via moment map to the little Weyl group of X.

Appeared in: Commentarii Mathematici Helvetici 70 (1995) 285-309.

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Last updated: March 10, 2004