Unitary $L^{p+}$-representations of almost automorphism groups

Antje Dabeler; Emilie Mai Elkiær; Maria Gerasimova; Tim de Laat

doi:10.5802/crmath.549

Research article - Representation theory

Unitary

L^{p +}

-representations of almost automorphism groups

Antje Dabeler ¹; Emilie Mai Elkiær ²; Maria Gerasimova ¹; Tim de Laat ¹

¹ University of Münster, Mathematical Institute, Einsteinstraße 62, 48149 Münster, Germany.
² Department of Mathematics, University of Oslo, Norway

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 245-249.

Abstract
Résumé

Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze–Stein property, and let $π$ be a unitary representation of $H$ . We show that the representation $\tilde{π}$ of $G$ induced from $π$ is an $L^{p +}$ -representation if and only if $π$ is an $L^{p +}$ -representation. We deduce the following consequence for a large natural class of almost automorphism groups $G$ of trees: For every $p \in (2, \infty)$ , the group $G$ has a unitary $L^{p +}$ -representation that is not an $L^{q +}$ -representation for any $q < p$ . This in particular applies to the Neretin groups.

Soit $G$ un groupe localement compact avec un sous-groupe ouvert $H$ ayant la propriété de Kunze–Stein, et soit $π$ une représentation unitaire de $H$ . Nous montrons que la représentation $\tilde{π}$ de $G$ induite par $π$ est une représentation $L^{p +}$ si et seulement si $π$ est une représentation $L^{p +}$ . Nous en déduisons la conséquence suivante pour une grande classe naturelle de groupes de presqu’automorphismes $G$ d’un arbre : pour tout $p \in (2, \infty)$ , le groupe $G$ a une représentation unitaire $L^{p +}$ qui n’est pas une représentation $L^{q +}$ pour tout $q < p$ . Ceci s’applique en particulier aux groupes de Neretin.

Received: 2023-04-27
Accepted: 2023-07-21
Published online: 2024-05-02

DOI: 10.5802/crmath.549

Author's affiliations:

Antje Dabeler ¹; Emilie Mai Elkiær ²; Maria Gerasimova ¹; Tim de Laat ¹

¹ University of Münster, Mathematical Institute, Einsteinstraße 62, 48149 Münster, Germany.
² Department of Mathematics, University of Oslo, Norway

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Antje Dabeler and Emilie Mai Elki{\ae}r and Maria Gerasimova and Tim de Laat},
     title = {Unitary $L^{p+}$-representations of almost automorphism groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {245--249},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.549},
     language = {en},
}

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Antje Dabeler; Emilie Mai Elkiær; Maria Gerasimova; Tim de Laat. Unitary $L^{p+}$-representations of almost automorphism groups. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 245-249. doi : 10.5802/crmath.549. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.549/

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