A surprizing fact we found recently is that:

 If SR/L is a
compact complex parallelizable manifold. S is the semisimple part
(R is the radical part) of SR and L is a cocompact discrete subgroup.
Then the the factors of S which act on R nontrivially are 
anisotropic A_{l} type groups.
<p> 
A first application is the classification
of compact complex homogeneous spaces. (Abstract
published on Mar. 20, 1998)
<p>
In the first part we deal with the case of compact complex 
homogeneous spaces with 1-step, i.e., the nilradical of the
parallelizable fiber of the Tits fibration is abelian. In the
second part we shall finish the classification of the general compact
complex homogeneous spaces. (Abstract published on Aug. 4, 1998)