Orbits of real forms in complex flag manifolds

Abstract

The authors study the structure and the CR geometry of the orbits $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a complex flag manifold $X = G/Q$. It is shown that any such orbit $M$ has a tower of fibrations over a canonically associated real flag manifold $M_e$ with fibers that are products of Euclidean complex spaces and open orbits in complex flag manifolds. This result is used to investigate some topological properties of $M$. For example, it is proved that the fundamental group $\pi_1(M)$ depends only on $M_e$ and on the conjugacy class of the maximally noncompact Cartan subgroups of the isotropy of the action of $G_0$ on $M$. In particular, the fundamental group of a closed orbit $M$ is isomorphic to that of $M_e$. Many other deep results about properties of the CR structure of the orbits and its invariants and about $G_0$-equivarant maps between orbits are obtained.