On homogeneous and symmetric CR manifolds

In this paper, some questions about CR homogeneous structures are studied. In particular, conditions for which an abstract CR structure (related to a Lie algebra) can be realized as a true homogeneous CR structure are given. A main tool is the Levi-Malʹtsev and the Jordan-Chevalley fibrations on homogeneous spaces. The authors study these fibrations for a CR homogeneous space. Examples and counterexamples are produced in this situation. The authors study in more detail the CR-homogeneous structures for two specific cases. 1. The case of a semi-simple Lie group of odd dimension. In the compact case, a classification in algebraic terms was given by J.-Y. Charbonnel and H. Ounaïes-Khalgui [J. Lie Theory 14 (2004), no. 1, 165--198; MR2040175 (2005b:22012)]. 2. The so-called CR-symmetric spaces introduced by W. Kaup and D. Zaitsev [Adv. Math. 149 (2000), no. 2, 145--181; MR1742704 (2000m:32044)] and which generalize to the CR situation the classical definition of Élie Cartan. At the end of the paper, the authors classify such structures for complete flag manifolds. This is done first for the classical simple groups and then for the exceptional groups. There are also results about the real algebraic case, in particular about the embedding problem in a complex space. The paper uses previous results given by Medori and Nacinovich.