Homological Finiteness of Representations of Almost Linear Nash Groups

Y. Bao ; Y. Chen

Journal of Lie theory, Tome 31 (2021) no. 4, pp. 1045-1053

Résumé

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.

Détail

Classification : 22E41
Mots-clés : Schwartz homology, tempered vector bundle, Schwartz sections, homological finiteness