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

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 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.