We study representations where 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 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 or their orthochronous component (where denotes the real, complex or quaternionic numbers) are Polish groups that are topologically simple but not abstractly simple.
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@article{AFST_2023_6_32_2_371_0,
author = {Bruno Duchesne},
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},
pages = {371--396},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {2},
year = {2023},
doi = {10.5802/afst.1740},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1740/}
}
TY - JOUR AU - Bruno Duchesne TI - Infinite dimensional representations of orthogonal groups of quadratic forms with finite index JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2023 SP - 371 EP - 396 VL - 32 IS - 2 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1740/ DO - 10.5802/afst.1740 LA - en ID - AFST_2023_6_32_2_371_0 ER -
%0 Journal Article %A Bruno Duchesne %T Infinite dimensional representations of orthogonal groups of quadratic forms with finite index %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2023 %P 371-396 %V 32 %N 2 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1740/ %R 10.5802/afst.1740 %G en %F AFST_2023_6_32_2_371_0
Bruno Duchesne. Infinite dimensional representations of orthogonal groups of quadratic forms with finite index. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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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