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

TL;DR

This paper completes the understanding of the spectral correspondence in rank one locally symmetric spaces by addressing exceptional parameters using vector valued Poisson transforms, extending previous work on hyperbolic surfaces and spaces.

Contribution

It introduces a method using vector valued Poisson transforms to handle exceptional spectral parameters in rank one locally symmetric spaces, completing the spectral correspondence program.

Findings

Treats exceptional spectral parameters with vector valued Poisson transforms.

Describes the nature of exceptional parameters in higher dimensions.

Completes the spectral correspondence for all rank one locally symmetric spaces.

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.