Here again, we let H = L2(Γ\G), δH = °L2(Γ\G), and V be the orthogonal complement of °H in H.

The space H is the Hilbert direct sum of the subspace Hm (m ∈ ℤ), so we already have a spectral decomposition of H with respect to C. The list in 15.8 shows that there are at most two inequivalent irreducible unitary representations of G with the same Casimir operator, so there is in principle not much difference between the decompositions with respect to C and to G. But we want to express the latter in terms of representations.

Lemma. Let (π, E) be a unitary representation of G. Assume the existence of a Dirac sequence {αn} (n ∈ ℕ) with compact operators π(αn). Then E is a Hilbert sum of irreducible representations, with finite multiplicities.

Proof. This is rather elementary and well-known. For the convenience of the reader we reproduce one proof, following 5.8 and 5.9 in [7].

(a) We show first that a closed G-invariant subspace W ≠ 0 contains a closed G-invariant subspace that is minimal among nonzero closed G-invariant subspaces. By 13.1 and 14.2, there exists a j such that π(αj)|w ≠ 0. Let c be a nonzero eigenvalue of π(αj) in W, and let M ≠ 0 be the corresponding (finite dimensional) eigenspace in W.