Real group orbits on flag manifolds

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.