Infinite dimensional representations of orthogonal groups of quadratic forms with finite index

Bruno Duchesne

doi:10.5802/afst.1740

Bruno Duchesne ¹

¹ Institut Élie Cartan, UMR 7502, Université de Lorraine et CNRS, Nancy, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 2, pp. 371-396.

Abstract

We study representations $G \to H$ where $G$ is either a simple Lie group with real rank at least 2 or an infinite dimensional orthogonal group of some quadratic form of finite index at least 2 and $H$ is such an orthogonal group as well. The real, complex and quaternionic cases are considered. Contrarily to the rank one case, we show that there is no exotic such representations and we classify these representations.

On the way, we make a detour and prove that the projective orthogonal groups ${PO}_{K} (p, \infty)$ or their orthochronous component (where $K$ denotes the real, complex or quaternionic numbers) are Polish groups that are topologically simple but not abstractly simple.

Received: 2019-09-04
Accepted: 2021-06-09
Published online: 2023-08-22

DOI: 10.5802/afst.1740

Author's affiliations:

Bruno Duchesne ¹

¹ Institut Élie Cartan, UMR 7502, Université de Lorraine et CNRS, Nancy, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Infinite dimensional representations of orthogonal groups of quadratic forms with finite index},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Bruno Duchesne. Infinite dimensional representations of orthogonal groups of quadratic forms with finite index. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 2, pp. 371-396. doi : 10.5802/afst.1740. https://afst.centre-mersenne.org/articles/10.5802/afst.1740/

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