Homological Finiteness of Representations of Almost Linear Nash Groups

Let $G$ be an almost linear Nash group, namely, a Nash group that admits a Nash homomorphism with finite kernel to some $\GL_k(\mathbb R)$. A smooth \Fre representation $V$ with moderate growth of $G$ is called homologically finite if the Schwartz homology $\oH_{i}^{\CS}(G;V)$ is finite dimensional for every $i\in\BZ$. We show that the space of Schwartz sections $\Gamma^{\varsigma}(X,\SE)$ of a tempered $G$-vector bundle $(X,\SE)$ is homologically finite as a representation of $G$, under some mild assumptions.