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

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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 general, the resulting representations turn out to be the relative discrete series representations for associated non-Riemannian symmetric spaces.

Citation

@article {AH23,
  AUTHOR = {Arends, Christian and Hilgert, Joachim},
  TITLE = {Spectral correspondences for rank one locally symmetric
    spaces: the case of exceptional parameters},
  JOURNAL = {J. \'Ec. polytech. Math.},
  FJOURNAL = {Journal de l'\'Ecole polytechnique. Math\'ematiques},
  VOLUME = {10},
  YEAR = {2023},
  PAGES = {335--403},
  URL = {https://doi.org/10.5802/jep.220},
}

Related material

• PhD dissertation (2023) - My PhD dissertation at Paderborn University is the basis for this paper and contains more detailed information.