In this chapter, we complete the proof of two main results outlined in Chapter 1 : spectral decomposition on standard pseudo-Riemannian locally homogeneous spaces and density of analytic eigenfunctions. To be precise, let us assume the existence of a reductive subgroup L acting properly and spherically on a reductive homogeneous space X = G/H. For any standard quotient XΓ = Γ\G/H defined by a torsion-free discrete subgroup Γ of L, we establish: Although our assumption permits the existence of a compact quotient XΓ, these theorems hold true regardless of compactness. The proof relies on the transfer map introduced in Chapter 5 , which is based on the natural fiber bundle structure of XΓ over the locally Riemannian symmetric space YΓ associated with L, with a compact fiber F. The sphericity assumption of L is crucial in establishing a connection between spectral analysis on the pseudo-Riemannian space XΓ and on the Riemannian space YΓ.