On a class of symmetric CR manifolds

Abstract

The classical notions of Riemannian and Hermitian symmetric spaces have recently been extended to CR manifolds by W. Kaup and the reviewer [Adv. Math. 149 (2000), no. 2, 145--181; MR1742704 (2000m:32044)]. Here, in order to contain natural examples such as spheres, one has to appropriately weaken the notion of symmetry by requiring that its differential at the reference point be the negative identity only in the complex tangent directions (and possibly other directions if the CR manifold is not of finite type). It was shown that examples of symmetric CR manifolds include spheres in ${\bf C}^n$Cn and, more generally, Shilov boundaries of bounded symmetric domains in their circular realizations.\ud The latter class of CR manifolds admits a natural generalization as minimal orbits in complex flag manifolds, which is the subject of the present paper. Consider a complex flag manifold $F=G^{\bf C}/Q$F=GC/Q, where $G^{\bf C}$GC is a connected semisimple complex Lie group with real form $G$G and $Q\subset G^{\bf C}$Q⊂GC is a parabolic subgroup. Let $M$M be the $G$G-orbit in $F$F of the minimal possible dimension (the minimal orbit), which is uniquely determined by this condition and is compact.\ud The authors begin by associating a complex flag manifold to every Levi-Tanaka algebra (the latter having been extensively studied by C. Medori and the second author in a series of papers) and showing that the minimal orbit $M$M is always a symmetric CR manifold in this case. A key step is the construction of the symmetries, based on the property of a semisimple Levi-Tanaka algebra that its (partial) complex structure $J$J is induced by an inner derivation corresponding to an element $\tilde J$J˜ of the algebra. The latter property is called the "$(J)$(J) property''. Then the symmetries are obtained from the action by the group element $\exp(\pi \tilde J)$exp(πJ˜) and its conjugations.\ud As a next step, the conclusion is extended to more general classes of complex flag manifolds having the $(J)$(J) property. Finally, a complete classification for fundamental minimal parabolic CR algebras having the $(J)$(J) property is given in terms of the so-called $\sigma$σ-diagrams

CR-Admissible $$\mathbb{{Z}}_{2}$$ -gradations and CR-symmetries

“…In [8] we proved the implications (b) ⇒ (c) and (b) ⇒ (a), and classified all parabolic minimal CR algebras satisfying (b). Moreover, the last assertions of Theorem 3.2 summarize results that are already proved there.…”

“…The endomorphisms ρ(A), for A ∈ g ∩ q, commute with J and hence ρ(g ∩ q) is a Lie algebra of complex endomorphisms of H. This J is called the partial complex structure of (g, q). According to [8], we say that (g, q) has Property (J) if there existsJ ∈ g ∩ q such that ρ(J) = J.…”

“…Proof As we already observed in [8], condition (C) means that : for each α ∈ ∩ R • the connected component of α in B ∩ R • is one of the following:…”

“…section 2).We can also assume (see [1,8]) that R + ⊂ Q, for the parabolic set of roots Q = {α | g α ⊂ q} associated to q.…”

“…In Theorem 8.1 of [8], we characterized property (J) by the requirement that the cross marked Satake diagram (S, ) of our fundamental parabolic minimal CR algebra (g, q) satisfies some properties, that we called (A), (B), (C) there. Thus the proof will consist in showing that a) implies that (A), (B), (C) are valid.…”