Journal of Lie Theory 31 (2021), No. 4, 1045--1053
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
[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.
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