Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle E on X. Let h be the Lie algebra of H. Let S(X, E) be the space of Schwartz sections of E. We prove that hS(X, E) is a closed subspace of S(X, E) of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let π be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants V _h → π _h is an isomorphism if and only if any linear h-invariant functional on V is continuous in the topology induced from π. The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.