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

الملخص EN

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 .