  We present necessary and sufficient conditions to have global hypoellipticity
and global solvability for a class of vector fields defined on a product of
compact Lie groups. In view of Greenfield's and Wallach's conjecture, about the
non-existence of globally hypoelliptic vector fields on compact manifolds
different from tori, we also investigate different notions of regularity weaker
than global hypoellipticity and describe completely the global hypoellipticity
and global solvability of zero-order perturbations of our vector fields. We
also present a class of vector fields with variable coefficients whose
operators can be reduced to a normal form, and we prove that the study of the
global properties of such operators is equivalent to the study of the
respective properties for their normal forms.
