
Journal of Lie Theory 19 (2009), No. 2, 267274 Copyright Heldermann Verlag 2009 Invariant Semisimple CR Structures on the Compact Lie Groups SU(n) and SO(p,R), 5 ≤ p ≤ 7 Hella OunaïesKhalgui Dép. de Mathématiques, Faculté des Sciences, Université Tunis ElManar, Campus Universitaire, 1060 Tunis, Tunisia o.khalgui@fst.rnu.tn Rupert W. T. Yu Dép. de Mathématiques, Université de Poitiers, Téléport 2  BP 30179, Blvd Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France yuyu@math.univpoitiers.fr [Abstractpdf] Let $G_{0}$ be a compact real Lie group of dimension $N$ and denote by $\g_{0}$ its Lie algebra. Recently J.Y. Charbonnel and the first author [Classification des structures CR invariantes pour les groupes de Lie compacts, Journal of Lie Theory 14 (2004) 165198] studied $G_{0}$invariant {\it CR} structures on $G_{0}$. Such a structure is defined by the fiber of the identity element of $G_{0}$ which is a Lie subalgebra $\h$ of the complexification $\g$ of $\g_{0}$, having trivial intersection with $\g_{0}$. If the dimension of the {\it CR} structure is maximal, that is $\left[N\over2\right]$, then Charbonnel and the first author showed that $\h$ is a solvable Lie algebra. In this note, we are interested in $G_{0}$invariant {\it CR} structures on $G_{0}$ which are defined by a semisimple Lie subalgebra and of maximal dimension. We distinguish two types of these {\it CR} structures which we shall call {\it CRSS} structure of type I and of type II. In the case of the group SU$(n)$, with $n\geq 3$, we show that there exists always a {\it CRSS} structure of type I, while in the case of SO$(p,\R)$, with $5\leq p\leq 7$, we show that a {\it CRSS} structure of type II exists. We obtain from these structures for each of these groups an almost global {\it CR} embedding into a finitedimensional complex vector space. Keywords: Compact Lie group, CauchyRiemann Structure, CRembedding. MSC: 22E99, 32V40, 57S15 [ Fulltextpdf (180 KB)] for subscribers only. 