Harmonic analysis of the

This is the fourth in a series of papers (the earlier ones are [Casselman:1989], [Casselman:1993] and [Casselman:1999]) intended to present eventually a new way of proving, among other things, the well known results of Chapter 7 of [Langlands:1977] on the completeness of the spectrum arising from cusp forms and Eisenstein series. That Langlands’ results have lasted for nearly 40 years without major improvements is testimony to their depth, but—to (mis)quote Peter Sarnak, who had some recent work of Joseph Bernstein’s in mind—it is time to reconsider the theory.