Quantization of cohomology in semi-simple Lie algebras

The space of realizations of a finite-dimensional Lie algebra by first order differential operators is naturally isomorphic to H^1 with coefficients in the module of functions. The condition that a realization admits a finite-dimensional invariant subspace of functions seems to act as a kind of quantization condition on this H^1. It was known that this quantization of cohomology holds for all realizations on 2-dimensional homogeneous spaces, but the extent to which quantization of cohomology is true in general was an open question. The present article presents the first known counter-examples to quantization of cohomology; it is shown that quantization can fail even if the Lie algebra is semi-simple, and even if the homogeneous space in question is compact. A explanation for the quantization phenomenon is given in the case of semi-simple Lie algebras. It is shown that the set of classes in H^1 that admit finite-dimensional invariant subspaces is a semigroup that lies inside a finitely-generated abelian group. In order for this abelian group be a discrete subset of H^1, i.e. in order for quantization to take place, some extra conditions on the isotropy subalgebra are required. Two different instances of such necessary conditions are presented.