Compact homogeneous CR manifolds (Q1608527)

In a previous paper [Res. Expo. Math. 25, 3-44 (2002; Zbl 1030.32025)] the authors studied and divided the space of compact, simply connected homogeneous CR manifolds having a compact transitive automorphism group into three disjoint classes. One of these three classes consists of so-called standard homogeneous CR manifolds, that is, homogeneous circle bundles over a flag manifold. In this paper, the authors make a more detailed classification. First of all, they prove that a non-standard homogeneous CR manifold is never CR equivalent to a standard CR manifold. Then they study all possibilities for two non-standard homogeneous CR manifolds to be CR equivalent. For the class of standard homogeneous CR manifolds, they give a criterion for the CR equivalence in terms of associated maximal compact automorphism groups.