The CR structure of minimal orbits in complex flag manifolds

Let $G$ be a real form of a complex semisimple Lie group $\widehat G$Gˆ, $P\subset\widehat G$P⊂Gˆ a parabolic subgroup and $V=\widehat G/P$V=Gˆ/P the corresponding flag manifold of $\widehat G$Gˆ. As was shown by J. A. Wolf in [Bull. Amer. Math. Soc. 75 (1969), 1121--1237; MR0251246 (40 #4477)], there exists exactly one $G$G-orbit $M$M on $V$V that is closed (and hence compact) in $V$V. This orbit is connected, of minimal dimension amongst the $G$G-orbits and it is naturally endowed with a $G$G-invariant CR structure, namely the one induced by the $\widehat G$Gˆ-invariant complex structure of $V$V. Let us call a homogeneous CR manifold which occurs as a closed orbit of this kind a minimal parabolic orbit. Observe that any minimal parabolic orbit $M$M is (up to equivalences) uniquely determined just by a pair $(G,P)$(G,P), where $G$G is a real form of a complex semisimple Lie group $\widehat G$Gˆ and $P$P is a parabolic subgroup of $\widehat G$Gˆ. The authors single out a set of properties satisfied by the pairs of Lie algebras $(\germ g={\rm Lie}(G),\germ p={\rm Lie}(P))$(g=Lie(G),p=Lie(P)) associated with minimal parabolic orbits and prove that, conversely, any pair satisfying such properties (called an effective minimal parabolic CR algebra) is associated with a minimal parabolic orbit in some flag manifold. After this, they show that for a fixed real semisimple Lie algebra $\germ g$g, the effective minimal parabolic CR algebras of the form $(\germ g,\germ p)$(g,p) are in one-to-one correspondence with a special class of cross-markings of the Satake diagram of $\germ g$g. Then, they study some topological properties of the homogeneous CR manifolds associated with effective minimal parabolic CR algebras and give complete characterizations of those, whose associated minimal parabolic orbits satisfy some special conditions on their CR structure. Such characterizations determine immediately new families of examples of CR manifolds that are either CR separable or of finite type or that satisfy nondegeneracy conditions of various kinds.