An External Approach to Unitary Representations

The principal ideas of harmonic analysis on a locally compact group G which is not necessarily compact or commutative were developed in the 1940s and early 1950s. In this theory, the role of the classical fundamental harmonics is played by the irreducible unitary representations of G. The set of all equivalence classes of such representations is denoted by Ĝ and is called the dual object of G or the unitary dual of G. Since the 1940s, an intensive study of the foundations of harmonic analysis on complex and real reductive groups has been in progress (for a definition of reductive groups, the reader may consult the appendix at the end of §2). The motivation for this development came from mathematical physics, differential equations, differential geometry, number theory, etc. Through the 1960s, progress in the direction of the Plancherel formula for real reductive groups was great, due mainly to HarishChandra’s monumental work, while at the same time, the unitary duals of only a few groups had been parametrized. With Mautner’s work [Ma], a study of harmonic analysis on reductive groups over other locally compact nondiscrete fields was started. We shall first describe such fields. In the sequel, a locally compact nondiscrete field will be called a local field. If we have a nondiscrete absolute value on the field Q of rational numbers, then it is equivalent either to the standard absolute value (and the completion is the field R of real numbers) or to a p-adic absolute value for some prime number p. For r ∈ Q write r = pa/b where α, a, and b are integers and neither a nor b are divisible by p. Then the p-adic absolute value of r is