The Spectrum of Kleinian Manifolds

We obtain the Plancherel theorem for L2(Γ\G), where G is a classical simple Lie group of real rank one and Γ⊂G is convex–cocompact discrete subgroup, and deduce its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. As the main tool, we develop a geometric version of scattering theory which, in particular, contains the meromorphic continuation of the Eisenstein series for this situation. The central role played by invariant distribution sections supported on the limit set is emphasized.