In Chapter 4 , we introduced a decomposition of joint L2-eigenfunctions of intrinsic differential operators on pseudo-Riemannian locally homogeneous spaces XΓ = Γ\G/H into type I and type II: type I eigenfunctions arise from distribution vectors of discrete series representations for X = G/H, while type II eigenfunctions are defined by taking an orthogonal complement in L2(XΓ). One difficulty with type II arises from the fact that its definition relies on the L2-inner product for XΓ, which differs from those for Γ\G or G/H because neither Γ nor H is compact, and thus the unitary representation theory of G cannot be applied directly. To overcome this difficulty, we consider a similar and easier concept for L-equivariant Hermitian vector bundles \(\mathcal{V}\) over the Riemannian locally symmetric space YΓ = Γ\Y associated with the subgroup L, by using Harish-Chandra’s discrete series representations for L. We demonstrate that the “transfer maps” preserve discrete spectrum of type I and of type II between L2(XΓ) and L2(YΓ, \(\mathcal{V}\) ) under the assumption that the reductive subgroup L acts properly and spherically on X. The results of this chapter are crucial in proving the existence of infinite discrete spectrum of type II in Chapter 10 .