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

Abstract

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle ${\mathcal {E}}$ on X. Let $\mathfrak {h}$ be the Lie algebra of H. Let $\mathcal {S}(X,{\mathcal {E}})$ be the space of Schwartz sections of ${\mathcal {E}}$. We prove that $\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})$ is a closed subspace of $\mathcal {S}(X,{\mathcal {E}})$ 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 $\pi$ be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants $V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}$ is an isomorphism if and only if any linear $\mathfrak {h}$-invariant functional on V is continuous in the topology induced from $\pi$. 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.