An external approach to unitary representations

Abstract

The principal ideas of harmonic analysis on a locally compact group G which is not necessarily compact or commutative, were developed in the 1940’s and early 1950’s. 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 1940’s, 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 the second section). The motivation for this development came from mathematical physics, differential equations, differential geometry, number theory, etc. Through the 1960’s, progress in the direction of the Plancherel formula for real reductive groups was great, due mainly to Harish-Chandra’s monumental work, while, at the same time, the unitary duals of only a few groups had been parametrized. With F. Mautner’s work [Ma], a study of harmonic analysis on reductive groups over other locally compact non-discrete fields was started. We shall first describe such fields. In the sequel, a locally compact non-discrete field will be called a local field. If we have a non-discrete 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 2 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 jrjp = p−α. A completion of Q with respect to the p-adic absolute value is denoted by Qp. It is called a field of p-adic numbers. Each finite dimensional extension F of Qp has a natural topology of a vector space over Qp. With this topology, F becomes a local field. The topology of F can be also introduced with an absolute value which is denoted by j jF (in the fifth section we shall fix a natural absolute value). The fields of real and complex numbers, together with the finite extensions of p-adic numbers, exhaust all local fields of characteristic zero up to isomorphisms ([We]). Let F be a finite field. Denote by F((X)) the field of formal power series over F. Elements of this field are series of the form f = P∞ n=k anX , an 2 F, for some integer k. Fix q > 1.