Spectral Correspondences for Rank One Locally Symmetric Spaces -- The Case of Exceptional Parameters

DOI10.5802/JEP.220arXiv2112.11073MaRDI QIDQ6386283

Christian Arends, Joachim Hilgert

Publication date: 21 December 2021

Abstract: 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 by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for general rank one spaces. Except for the case of hyperbolic surfaces a countable set of exceptional spectral parameters always left 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 $mathrmSL(2,mathbbR)$. In higher dimensions the situation is more complicated, but can be described completely.

Mathematics Subject Classification ID

Semisimple Lie groups and their representations (22E46) Discrete subgroups of Lie groups (22E40) Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. (37C30)

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