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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1778-3569 |