Due to the lack of a comprehensive theory in pseudo-Riemannian geometry, it is unclear whether there exists an infinite L2 spectrum for the Laplacian on compact pseudo-Riemannian manifolds in general. In this chapter, we establish the existence of an infinite L2 spectrum for intrinsic differential operators on each compact standard locally homogeneous space XΓ = Γ\G/H that is defined by a proper spherical action of a reductive subgroup L and a cocompact discrete subgroup Γ of L. Furthermore, we extend the results to the case where Γ is an arithmetic subgroup of L. This includes the situation where XΓ is noncompact but of finite volume. We actually prove the existence of an infinite spectrum of type II, as introduced in Chapter 4 . Our proof relies on the transfer maps of Chapter 7 , which connect spectrum in pseudo-Riemannian locally homogeneous spaces and in vector bundles over Riemannian locally symmetric spaces.