
Journal of Lie Theory 31 (2021), No. 4, 10451053 Copyright 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 [Abstractpdf] 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. [ Fulltextpdf (108 KB)] for subscribers only. 