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 ${\text{GL}}_k(\mathbb{R})$. A smooth \Fre representation $V$ with moderate growth of $G$ is called homologically finite if the Schwartz homology ${\text{oH}}_i^{\text{CS}}(G;V)$ is finite dimensional for every $i\in \text{BZ}$. We show that the space of Schwartz sections ${\mathrm{\Gamma }}^{\varsigma }(X,\text{SE})$ of a tempered $G$-vector bundle $(X,\text{SE})$ is homologically finite as a representation of $G$, under some mild assumptions.