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

C. Arends, J. Hilgert, ArXiv:2112.11073 (2021).

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Author
Arends, ChristianLibreCat; Hilgert, Joachim
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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 $\mathrm{SL}(2,\mathbb R)$. In higher dimensions the situation is more complicated, but can be described completely.
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Journal Title
arXiv:2112.11073
Page
60
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### Cite this

Arends C, Hilgert J. Spectral Correspondences for Rank One Locally Symmetric Spaces -- The  Case of Exceptional Parameters. arXiv:211211073. Published online 2021.
Arends, C., & Hilgert, J. (2021). Spectral Correspondences for Rank One Locally Symmetric Spaces -- The  Case of Exceptional Parameters. In arXiv:2112.11073.
@article{Arends_Hilgert_2021, title={Spectral Correspondences for Rank One Locally Symmetric Spaces -- The  Case of Exceptional Parameters}, journal={arXiv:2112.11073}, author={Arends, Christian and Hilgert, Joachim}, year={2021} }
Arends, Christian, and Joachim Hilgert. “Spectral Correspondences for Rank One Locally Symmetric Spaces -- The  Case of Exceptional Parameters.” ArXiv:2112.11073, 2021.
C. Arends and J. Hilgert, “Spectral Correspondences for Rank One Locally Symmetric Spaces -- The  Case of Exceptional Parameters,” arXiv:2112.11073. 2021.
Arends, Christian, and Joachim Hilgert. “Spectral Correspondences for Rank One Locally Symmetric Spaces -- The  Case of Exceptional Parameters.” ArXiv:2112.11073, 2021.

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arXiv 2112.11073