The Problem of Classifying Automorphic Representations of Classical Groups

In this article we shall give an elementary introduction to an important problem in representation theory. The problem is to relate the automorphic representations of classical groups to those of the general linear group. Thanks to the work of a number of people over the past twenty-five years, the automorphic representation theory of GL(n) is in pretty good shape. The theory for GL(n) now includes a good understanding of the analytic properties of Rankin-Selberg L-functions, the classification of the discrete spectrum, and cyclic base change. One would like to establish similar things for classical groups. The goal would be an explicit comparison between the automorphic spectra of classical groups and GL(n) through the appropriate trace formulas. There are still obstacles to be overcome. However with the progress of recent years, there is also reason to be optimistic. We shall not discuss the techniques here. Nor will we consider the possible applications. Our modest aim is to introduce the problem itself, in a form that might be accessible to a nonspecialist. In the process we shall review some of the basic constructions and conjectures of Langlands that underlie the theory of automorphic representations.