Unitary $L^{p+}$-representations of almost automorphism groups
Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze–Stein property, and let $\pi $ be a unitary representation of $H$. We show that the representation $\widetilde{\pi }$ of $G$ induced from $\pi $ is an $L^{p+}$-representation if and only if $\pi $ is an $L^{p+}$-representati...
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.549/ |
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| author | Dabeler, Antje Mai Elkiær, Emilie Gerasimova, Maria de Laat, Tim |
| author_facet | Dabeler, Antje Mai Elkiær, Emilie Gerasimova, Maria de Laat, Tim |
| author_sort | Dabeler, Antje |
| collection | DOAJ |
| description | Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze–Stein property, and let $\pi $ be a unitary representation of $H$. We show that the representation $\widetilde{\pi }$ of $G$ induced from $\pi $ is an $L^{p+}$-representation if and only if $\pi $ 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. |
| format | Article |
| id | doaj-art-488e9faa5c7748619294263939437baa |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-488e9faa5c7748619294263939437baa2025-02-07T11:19:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G324524910.5802/crmath.54910.5802/crmath.549Unitary $L^{p+}$-representations of almost automorphism groupsDabeler, Antje0Mai Elkiær, Emilie1Gerasimova, Maria2de Laat, Tim3University of Münster, Mathematical Institute, Einsteinstraße 62, 48149 Münster, Germany.Department of Mathematics, University of Oslo, NorwayUniversity of Münster, Mathematical Institute, Einsteinstraße 62, 48149 Münster, Germany.University of Münster, Mathematical Institute, Einsteinstraße 62, 48149 Münster, Germany.Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze–Stein property, and let $\pi $ be a unitary representation of $H$. We show that the representation $\widetilde{\pi }$ of $G$ induced from $\pi $ is an $L^{p+}$-representation if and only if $\pi $ 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.549/ |
| spellingShingle | Dabeler, Antje Mai Elkiær, Emilie Gerasimova, Maria de Laat, Tim Unitary $L^{p+}$-representations of almost automorphism groups Comptes Rendus. Mathématique |
| title | Unitary $L^{p+}$-representations of almost automorphism groups |
| title_full | Unitary $L^{p+}$-representations of almost automorphism groups |
| title_fullStr | Unitary $L^{p+}$-representations of almost automorphism groups |
| title_full_unstemmed | Unitary $L^{p+}$-representations of almost automorphism groups |
| title_short | Unitary $L^{p+}$-representations of almost automorphism groups |
| title_sort | unitary l p representations of almost automorphism groups |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.549/ |
| work_keys_str_mv | AT dabelerantje unitarylprepresentationsofalmostautomorphismgroups AT maielkiæremilie unitarylprepresentationsofalmostautomorphismgroups AT gerasimovamaria unitarylprepresentationsofalmostautomorphismgroups AT delaattim unitarylprepresentationsofalmostautomorphismgroups |