Foreword

In 1997 the annual instructional conference of the International Centre for

Mathematical Sciences in Edinburgh was devoted to the representation theory

of semisimple groups, to automorphic forms, and to the relations between these

subjects. It was organized by T. N. Bailey, L. Clozel, M. Duflo, and A. W. Knapp.

The two-week meeting began with a rapid summary of basic theory and concluded

with two lectures by Robert Langlands, returning from the award of the Wolf

Prize. In between, fifteen other world experts gave courses of two to five lectures.

There were close to one hundred participants, largely from Western Europe and

North America, but also from Eastern Europe, Japan, and the Developing World.

Funding for the conference was provided by the European Commission and the

Engineering and Physical Sciences Research Council of the United Kingdom.

The papers in this volume consist of slightly expanded versions of the lectures,

with some minor rearrangements. An exception is the paper by James Arthur,

which is a version of a lecture given at a later conference. All papers were re-

ceived before May 1, 1997, and were refereed. The papers are intended to provide

overviews of the topics they address, and the authors have supplied extensive bib-

liographies to guide the reader who wants more detail. The editors hope that the

papers will serve partly as guides to the literature and that readers at any level will

be able to get an outline of new ideas that they will be able to fill in by following

the references. As is true in the mathematical literature generally, different authors

use slightly different definitions and notation. A global index at the end of the

volume may help the reader reconcile the differences.

The aim of the conference was to provide an intensive treatment of representation

theory for two purposes: One was to help analysts to make systematic use of

Lie groups in work on harmonic analysis, differential equations, and mathematical

physics, and the other was to treat for number theorists the representation-theoretic

input to Wiles's proof of Fermat's Last Theorem.

It is tempting to think of the lectures and papers as consisting of a common core

and two more advanced partsâ€”one going in the direction of analysis on semisimple

groups G and semisimple symmetric spaces G/H and the other going in the direc-

tion of properties of cusp and automorphic forms, their associated number theory,

and properties of G/T for arithmetic subgroups T. But the editors have resisted the

temptation to organize the proceedings in this fashion, because this would ignore

the important historical interplay between the two subjects.

This interplay goes in both directions, as evidenced in many of the papers. The

Langlands conjecture on discrete series of G, which is discussed in Schmid's paper,

came about when Langlands took a known theorem about G/r, put T = 1, and

made a heuristic calculation about what should happen. The standard intertwining

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