@johncarlosbaez Exactly! One place where we see a finite dimensional space of (G)-invariant forms is on reductive homogeneous manifolds. If (M=G/H) is a reductive homogeneous manifold, then one has a decomposition (\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{b}) where (\mathfrak{b}) is merely a subspace of (\mathfrak{g}) which is also invariant under the adjoint action of (H). The space of (G)-invariant (k)-forms can then be identified with a subspace of (\wedge^k\mathfrak{b}^\ast) where (\mathfrak{b}^\ast) is the dual of (\mathfrak{b}). So this is more general than Lie groups and everything can still be reduced to algebra which is very nice.