Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters

Christian Arends; Joachim Hilgert

doi:10.5802/jep.220

Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters
[Correspondances spectrales pour les espaces localement symétriques de rang

1

: le cas des paramètres exceptionnels]

Christian Arends ¹ ; Joachim Hilgert ¹

¹ Institut für Mathematik, Universität Paderborn Warburger Str. 100, 33098 Paderborn, Germany

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 335-403.

Résumé
Abstract

Dans cet article, nous complétons le programme sur la correspondance entre le spectre du laplacien des espaces localement symétriques compacts de rang $1$ et la première bande de résonances de Ruelle-Pollicott de leur flot géodésique sur le fibré en sphères. Ce programme a débuté dans [FF03] par Flaminio et Forni pour les surfaces hyperboliques, poursuivi dans [DFG15] pour les espaces hyperboliques réels et dans [GHW21] pour les espaces généraux de rang $1$ . À l’exception du cas des surfaces hyperboliques (voir aussi [GHW18]), un ensemble dénombrable de paramètres spectraux exceptionnels n’a pas été traité, la raison étant que les transformées de Poisson correspondantes ne sont ni injectives ni surjectives. Nous utilisons des transformées de Poisson à valeurs vectorielles pour traiter ces paramètres spectraux exceptionnels. Pour les surfaces, les paramètres spectraux exceptionnels conduisent à des représentations en série discrète de $SL (2, ℝ)$ (voir [FF03, GHW18]). En général, les représentations que l’on obtient s’avèrent être les représentations en série discrète relatives pour les espaces symétriques non riemanniens associés.

In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started in [FF03] by Flaminio and Forni for hyperbolic surfaces, continued in [DFG15] for real hyperbolic spaces and in [GHW21] for general rank one spaces. Except for the case of hyperbolic surfaces (see also [GHW18]) a countable set of exceptional spectral parameters always remained untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of $SL (2, ℝ)$ (see [FF03, GHW18]). In general, the resulting representations turn out to be the relative discrete series representations for associated non-Riemannian symmetric spaces.

Reçu le : 2022-03-09
Accepté le : 2023-01-23
Publié le : 2023-02-22

DOI : 10.5802/jep.220

Classification : 22E46, 22E40, 37C30, 37C79, 37D40
Keywords: Ruelle resonances, Poisson transforms, locally symmetric spaces, principal series representations
Mot clés : Résonances de Ruelle, transformées de Poisson, espaces localement symétriques, représentations en série principale

Affiliations des auteurs :

Christian Arends ¹ ; Joachim Hilgert ¹

¹ Institut für Mathematik, Universität Paderborn Warburger Str. 100, 33098 Paderborn, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Christian Arends; Joachim Hilgert. Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 335-403. doi : 10.5802/jep.220. https://jep.centre-mersenne.org/articles/10.5802/jep.220/

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