Journal of Lie Theory

Homological finiteness of principal series representations of $\mathrm{GL}_n(\mathbb{R})$
Chen, Yangyang1; Yang, Jeffrey2
1School of Mathematics and Data Science Jiangnan University Wuxi, 214122, China
2Shenzhen College of International Education Shenzhen, 518043, China
Journal of Lie Theory, Online first, pp. 1-13

Let $G=\mathrm{GL}_n(\mathbb{R})$ be the general linear group over $\mathbb{R}$ and let $B$ denote the Borel subgroup of $G$ consisting of nonsingular upper triangular matrices. For every principal series representation $\pi $ of $G$, we show that the Schwartz homology $\mathrm{H}_i(B;\pi )$ is finite-dimensional for every $i\in \mathbb{Z}$. The proof is based on a simple computation modulo the theory of Nash manifolds. Moreover, by applying Casselman’s subrepresentation theorem, we show that the conclusion still holds with the principal series representation $\pi $ replaced by a Casselman–Wallach representation.

Received:
Accepted:
DOI: 10.5802/jolt.1428
Classification: 22E46
Keywords: Homological finiteness, Principal series representations, Schwartz sections, General linear groups

Chen, Yangyang  1 ; Yang, Jeffrey  2

1 School of Mathematics and Data Science Jiangnan University Wuxi, 214122, China
2 Shenzhen College of International Education Shenzhen, 518043, China 
Chen, Yangyang; Yang, Jeffrey. Homological finiteness of principal series representations of $\mathrm{GL}_n(\mathbb{R})$. Journal of Lie Theory, Online first, pp. 1-13
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