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

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

Comptes Rendus. Mathématique, Tome 362 (2024) no. G3, pp. 245-249

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 $\tilde{\pi }$ 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.

Soit $G$ un groupe localement compact avec un sous-groupe ouvert $H$ ayant la propriété de Kunze–Stein, et soit $\pi$ une représentation unitaire de $H$. Nous montrons que la représentation $\tilde{\pi }$ de $G$ induite par $\pi$ est une représentation $L^{p+}$ si et seulement si $\pi$ 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.

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