Parametrization of the orbits of the real forms SU (p, q) and SO(p, q) in Grassmannian

Abstract

Let G  be a complex semi-simple Lie group with real form G0. Let Z = G/P be identified with Gr(k, n), the Grassmannian of k planes in Cn. Equivalently, P is a maximal  parabolic subgroup defined by the dimension sequence (k, n − k). Consider the action of G0  on the Grassmannian Gr(k, n). It is known that G0 has only finitely many orbits in G/P and therefore it has a unique closed orbit and at least one open orbit ([2],[6]). In this paper we will prove that the G0-orbits in Gr(k, n) are parameterized by signature, where G0 is SU (p, q) and SO(p, q) a real form of SL(n, C) and SO(p, q) respectively .