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].