The spectral decomposition ends this description of some basic material on the analytic theory of automorphic forms on SL2(ℝ). This is, however, only the starting point of the theory. We now add some comments on other developments, mainly for orientation and to indicate some literature for further study, without aiming at completeness.

Let A(s, m) be the space of automorphic forms of right K-type m that are eigenfunctions of C with eigenvalue (s2 - 1)/2. By definition, this space is the orthogonal direct sum of the subspace °A(s, m) of cusp forms and of its orthogonal complement A1(s, m). In Section 12, we obtained some information on the latter: its dimension is equal to the number of cusps or neat cusps; it is generated by Eisenstein series E(r, m) holomorphic at s and suitable limits of Eisenstein series. Although this description is not quite exhaustive, depending notably on the poles of Eisenstein series, it is quite substantial, so that the main remaining issue is the determination of the cusp forms. Note that, in the cocompact case, A1(s, m) = {0} and so nothing has been achieved toward the description of A(s, m).

By 16.2, the cuspidal spectrum °L2(Γ\G) is a Hilbert direct sum of irreducible unitary G-modules π ∈ Ĝ with finite multiplicities, say m(π, Γ). We know the K-types of all irreducible unitary representations of G (see §15), and they all have multiplicity 1 (a very special property of SL2(ℝ)).