  We are interested in global properties of systems of left-invariant
differential operators on compact Lie groups: regularity properties, properties
on the closedness of the range and finite dimensionality of their cohomology
spaces, when acting on various function spaces e.g. smooth, analytic and
Gevrey. Extending the methods of Greenfield and Wallach (1973) to systems, we
obtain abstract characterizations for these properties and use them to derive
some generalizations of results due to Greenfield (1972), Greenfield and
Wallach (1972), as well as global versions of a result of Caetano and Cordaro
(2011) for involutive structures.
