On the spectral theory of groups  of affine transformations  of compact nilmanifolds

Bekka, Bachir; Guivarc'h, Yves

doi:10.24033/asens.2253

On the spectral theory of groups of affine transformations of compact nilmanifolds
[Sur la théorie spectrale des groupes de transformations affines des nilvariétés compactes]

Bekka, Bachir ; Guivarc'h, Yves

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 607-645.

Résumé
Abstract

Soit $N$ un groupe de Lie nilpotent, connexe et simplement connexe; soient $Λ$ un réseau dans $N$ et $Λ ∖ N$ la nilvariété correspondante. Nous donnons une caractérisation des sous-groupes dénombrables du groupe $Aff (Λ ∖ N)$ des transformations affines de $Λ ∖ N$ dont l'action sur $L^{2} (Λ ∖ N)$ possède un trou spectral : ce sont les groupes $H$ pour lesquels le tore quotient maximal $T$ de $Λ ∖ N$ ne possède aucun sous-tore propre et $H$ -invariant $S$ tel que la projection de $H$ sur $Aut (T / S)$ soit un groupe virtuellement abélien.

Les outils principaux de la preuve sont la théorie de Kirillov des représentations unitaires des groupes de Lie nilpotents et l'étude du comportement asymptotique des coefficients matriciels de la représentation métaplectique du groupe symplectique qui permettent de ramener le cas général à celui des tores dont l'étude est préalablement menée. Nos méthodes montrent que l'action de $H \subset Aff (Λ ∖ N)$ sur $Λ ∖ N$ est ergodique (ou celle de $H \subset Aut (Λ ∖ N)$ sur $Λ ∖ N$ est fortement mélangeante) si et seulement si l'action induite de $H$ sur $T$ possède la même propriété.

Let $N$ be a connected and simply connected nilpotent Lie group, $Λ$ a lattice in $N$ , and $Λ ∖ N$ the corresponding nilmanifold. We characterize the countable subgroups of the group $Aff (Λ ∖ N)$ of affine transformations of $Λ ∖ N$ whose action on $L^{2} (Λ ∖ N)$ has a spectral gap: these are the groups $H$ for which there exists no proper $H$ -invariant subtorus $S$ of the maximal torus factor $T$ of $Λ ∖ N$ such that the projection of $H$ on $Aut (T / S)$ is a virtually abelian group.

The result is first established when $Λ ∖ N$ is a torus. The problem for a general nilmanifold is reduced to the torus case, using Kirillov's theory of unitary representations of nilpotent Lie groups and decay properties of the metaplectic representation of the symplectic group. Our methods show that the action of $H \subset Aff (Λ ∖ N)$ on $Λ ∖ N$ is ergodic (or the action of $H \subset Aut (Λ ∖ N)$ on $Λ ∖ N$ is strongly mixing) if and only if the corresponding action of $H$ on $T$ has the same property.

Publié le : 2015-09-24

MR Zbl

DOI : 10.24033/asens.2253

Classification : 37A05, 22F30, 60B15, 60G50
Keywords: Nilmanifolds, groups of affine transformations, spectral gap, strong ergodicity, random walks on homogeneous spaces, metaplectic representation.
Mot clés : Nilvariétés, groupes de transformations affines, trou spectral, ergodicité forte, marches aléatoires sur les espaces homogènes, représentation métaplectique.

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     author = {Bekka, Bachir and Guivarc'h, Yves},
     title = {On the spectral theory of groups  of affine transformations  of compact nilmanifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {607--645},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {3},
     year = {2015},
     doi = {10.24033/asens.2253},
     mrnumber = {3377054},
     zbl = {1335.22009},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2253/}
}

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Bekka, Bachir; Guivarc'h, Yves. On the spectral theory of groups  of affine transformations  of compact nilmanifolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 607-645. doi : 10.24033/asens.2253. https://www.numdam.org/articles/10.24033/asens.2253/

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