Orbits of real forms in complex flag manifolds

TL;DR

This paper analyzes the geometric and topological properties of orbits of real forms in complex flag manifolds, focusing on CR geometry, finite type conditions, and Levi nondegeneracy, with graphical and fibration-based characterizations.

Contribution

It characterizes finite type and Levi nondegenerate orbits of real forms in complex flag manifolds using graphical diagrams and constructs fibrations to compute their fundamental groups.

Findings

Characterization of finite type orbits

Graphical diagrams for orbit properties

Fibrations for fundamental group computation

Abstract

We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semisimple Lie group G in a complex flag manifold G/Q. In particular we characterize those that are of finite type and satisfy some Levi nondegeneracy conditions. These properties are also graphically described by attaching to them some cross-marked diagrams that generalize those for minimal orbits that we introduced in a previous paper. By constructing canonical fibrations over real flag manifolds, with simply connected complex fibers, we are also able to compute their fundamental group.