Real group orbits on flag manifolds

TL;DR

This survey explores the action of real forms of complex semisimple Lie groups on flag manifolds, highlighting key theorems, new proofs, and the relationship between cycle spaces and crowns of symmetric spaces.

Contribution

It provides a new proof for the converse of Wolf's finiteness theorem for real forms of inner type and discusses the cycle spaces and crowns of symmetric spaces.

Findings

At least one orbit is open on flag manifolds.

The homogeneous space is a flag manifold if a real form has an open orbit.

Cycle space often coincides with the crown of a symmetric space.

Abstract

In this survey, we gather together various results on the action of a real form of a complex semisimple Lie group on its flag manifolds. We start with the finiteness theorem of J.Wolf implying that at least one of the orbits is open. We give a new proof of the converse statement for real forms of inner type, essentially due to F.M.Malyshev. Namely, if a real semisimple Lie group of inner type has an open orbit on an algebraic homogeneous space of the complexified group then the homogeneous space is a flag manifold. To prove this, we recall, partly with proofs, some results of A.L.Onishchik on the factorizations of reductive groups. Finally, we discuss the cycle spaces of open orbits and define the crown of a symmetric space of non-compact type. With some exceptions, the cycle space agrees with the crown. We sketch a complex analytic proof of this result, due to G.Fels, A.Huckleberry and J.Wolf.