We first review some notions and facts about continuous representations of

in a locally complete topological vector space V. In fact, all we need is the representation by right translations of G in C∞(G), L2(G), C∞(H\G), or L2(H\G) (H closed unimodular subgroup, mainly F). Much of this has been encountered earlier, implicitly or explicitly. All this is valid in a much more general framework (see e.g. [7]).

A continuous representation (π, V) of G into V is a homomorphism of G into the group of automorphisms of V that is continuous; in other words, the map (g, v) ↦ π(g).v is a continuous map of G × V into V. If V is a Hilbert space then it is said to be i>unitary if π (g) (g ∈ G) leaves the scalar product of V invariant. Then the operator norm ∥π(g)∥ is uniformly bounded (by 1), and it is known that continuity already follows from separate continuity:

(1) for every v ∈ V, the map g ↦ π(g).v of G into V is continuous (see e.g. [7, §3] or [10, §VIII.1]).

The assumption “locally complete” is made to ensure that if α ∈ Cc(G) then ∫G α(x)π(x).v dx converges to an element of V. More generally, ∫ µ(x)π(x).v converges if µ is a compactly supported measure on G (see an important example in 14.4).

Our main prerequisites may be listed as follows.

1.1Elementary theory of Lie groups and Lie algebras, and the interpretation of the elements of the Lie algebra as differential operators on the group or its coset spaces. This will be used only for SL2(ℝ), its Lie subgroups, and the upper halfplane. The book by Warner [58] is more than sufficient for our needs, except for some facts on Haar measures (2.9, 10.9).

1.2The regularity theorem for elliptic operators (see the remark in 2.13 for references).

1.3Some functional analysis, mainly about operators on Hilbert spaces. For the sake of definiteness, I have used two basic textbooks ([46] and [51]) and have given at least one precise reference for every theorem used. But this material is standard and the reader is likely to find what is needed in his (or her) favorite book on functional analysis. The demands will increase as we go along, and the material will be briefly reviewed before it is needed. Such review is mainly intended to refresh memory, fix notation, and give references, not to be a full-fledged introduction.

1.4Infinite dimensional representations of G. We shall review what we need in Sections 14 and 15 and also give a few proofs, but mostly refer to the literature. An essentially self-contained discussion of some of the results on SL2(ℝ) stated in Sections 2 and 14 is contained in the first part of [40].

A more accurate title would be: Introduction to some aspects of the analytic theory of automorphic forms on SL2(ℝ) and the upper half-plane X. Originally, automorphic forms were holomorphic or meromorphic functions on X satisfying certain conditions with respect to a discrete group Γ of automorphisms of X, usually with fundamental domain of finite (hyperbolic) area. Later on, H. Maass – and then A. Selberg and W. Roelcke – dropped the assumption of holomorphicity, requiring instead that the functions under consideration be eigenfunctions of the Laplace–Beltrami operator. In the 1950s it was realized (in more general cases) – initially by I. M. Gelfand and S. V. Fomin, and then by Harish-Chandra – that the automorphic forms (holomorphic or not) could be equivalently viewed as functions on Γ\SL2(ℝ) satisfying certain conditions familiar in the theory of infinite dimensional representations of semisimple Lie groups. This led to a new outlook, where the Laplace–Beltrami operator is replaced by the Casimir operator and the theory of automorphic forms becomes closely related to harmonic analysis on Γ\SL2(ℝ). This is the point of view adopted in this presentation. However, in order to limit the prerequisites, no knowledge of representation theory is assumed until the last sections, a main purpose of which is precisely to make this connection explicit. A fundamental role is played throughout by a theorem stating that a function on SL2(ℝ) satisfying certain assumptions (Z-finite and K-finite) is fixed under convolution by some smooth function with arbitrarily small compact support around the identity element (2.14).

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)).