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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

Yangyang Chen
School of Sciences, Jiangnan University, Wuxi, P. R. China


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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