 
Summary: Centre de Recherches Mathematiques
CRM Proceedings and Lecture Notes
Volume 11, 1997
The Problem of Classifying Automorphic Representations of
Classical Groups
James Arthur
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 twentyfive 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 RankinSelberg Lfunctions,
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
