Abstract: Let G be a reductive Nash group, acting on a Nash manifold X. Let Z be a G -stable closed Nash submanifold of X and denote by $U$ the complement of Z in X. Let $\chi$ be a character of G and denote by g the complexified Lie algebra of G. We give a sufficient condition for the natural linear map $H_{k}(g, S(U)\otimes\chi)\rightarrow H_{k}(g, S(X)\otimes\chi)$ between the Lie algebra homologies of Schwartz functions to be an isomorphism. For k=0, by considering the dual, we obtain the automatic extensions of $g$-invariant (twisted by -$\chi$) Schwartz distributions.

Key words: Schwartz functions, Lie algebra homology, Hausdorffness, Schwartz distributions, automatic extensions