Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture (Q303798)

Let \(H\) be a real algebraic group acting on a real algebraic manifold \(X\) and \(h\) be the Lie algebra of \(H\). Let \(\mathcal{E}\) be a real algebraic bundle on \(X\) and \(S(X,\mathcal{E})\) be the space of Schwartz sections of \(\mathcal{E}\). The main result of this article is as follows. Suppose that the number of \(H\)-orbits in \(X\) is finite and let \(\chi\) be a tempered character of \(H\). Then \(H_0(h,S(X, \mathcal{E}) \otimes \chi)\) is separated and finite-dimensional. In other words, the subspace \(hS(X,\mathcal{E}) \subset S(X,\mathcal{E})\) (where the action of \(h\) is twisted by \(\chi\)) is closed and has finite codimension. An application of this result in the case when \(H\) is a real spherical subgroup of a real reductive group \(G\) is given. An equivalence is proved of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology.