Homological Finiteness of Representations of Almost Linear Nash Groups

Y. Bao; Y. Chen

doi:10.5802/jolt.1210

Y. Bao; Y. Chen

Journal of Lie Theory, Volume 31 (2021) no. 4, pp. 1045-1053

Abstract

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.

arXiv Zbl

DOI: 10.5802/jolt.1210

Classification: 22E41
Keywords: Schwartz homology, tempered vector bundle, Schwartz sections, homological finiteness

@article{JOLT_2021_31_4_a10,
     author = {Y. Bao and Y. Chen},
     title = {Homological {Finiteness} of {Representations} of {Almost} {Linear} {Nash} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {1045--1053},
     year = {2021},
     volume = {31},
     number = {4},
     doi = {10.5802/jolt.1210},
     zbl = {1492.22011},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1210/}
}

TY  - JOUR
AU  - Y. Bao
AU  - Y. Chen
TI  - Homological Finiteness of Representations of Almost Linear Nash Groups
JO  - Journal of Lie Theory
PY  - 2021
SP  - 1045
EP  - 1053
VL  - 31
IS  - 4
UR  - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1210/
DO  - 10.5802/jolt.1210
ID  - JOLT_2021_31_4_a10
ER  -

%0 Journal Article
%A Y. Bao
%A Y. Chen
%T Homological Finiteness of Representations of Almost Linear Nash Groups
%J Journal of Lie Theory
%D 2021
%P 1045-1053
%V 31
%N 4
%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.1210/
%R 10.5802/jolt.1210
%F JOLT_2021_31_4_a10

Y. Bao; Y. Chen. Homological Finiteness of Representations of Almost Linear Nash Groups. Journal of Lie Theory, Volume 31 (2021) no. 4, pp. 1045-1053. doi: 10.5802/jolt.1210

Cited by Sources: