Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 31 (2021), No. 4, 1045--1053Copyright Heldermann Verlag 2021 Homological Finiteness of Representations of Almost Linear Nash Groups Yixin Bao School of Sciences, Harbin Institute of Technology, Shenzhen, P. R. China mabaoyixin1984@163.com Yangyang Chen School of Sciences, Jiangnan University, Wuxi, P. R. China 8202007345@jiangnan.edu.cn [Abstract-pdf] Let $G$ be an almost linear Nash group, namely, a Nash group that admits a Nash homomorphism with finite kernel to some ${\mathrm GL}_k(\mathbb R)$. A smooth Fr\'{e}chet representation $V$ with moderate growth of $G$ is called homologically finite if the Schwartz homology ${\mathrm H}_{i}^{\mathcal{S}}(G;V)$ is finite dimensional for every $i\in{\mathbb Z}$. We show that the space of Schwartz sections $\Gamma^{\varsigma}(X,{\mathrm E})$ of a tempered $G$-vector bundle $(X,{\mathrm E})$ is homologically finite as a representation of $G$, under some mild assumptions. Keywords: Schwartz homology, tempered vector bundle, Schwartz sections, homological finiteness. MSC: 22E41. [ Fulltext-pdf  (108  KB)] for subscribers only.