Journal of Lie Theory

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) 165--198] 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 finite-dimensional complex vector space.