Proceedings of Symposia in Pure Mathematics

Volume 68, 2000

Harish-Chandra, His Work, and its Legacy

V. S. Varadarajan

ABSTRACT. Starting from around the late 1940s and reaching into the early

1980s, Harish-Chandra created, almost single-handedly, the theory of repre-

sentations of and harmonic analysis on semisimple Lie groups and their homo-

geneous spaces. This report, which opens a retrospective of his work and its

influence, attempts to sketch briefly for the benefit of a wider mathematical

audience as well as younger mathematicians coming into this field, the outlines

of his work and the main ideas that inform it, and to create at the same time,

by some personal reminiscences, a portrait of a compelling personality.

1. Introduction

If Harish-Chandra were alive today, he would be 75 and I am sure, would not

only be very pleased with what is going on in his favorite part of mathematics,

but would also have many profound and insightful things to say about them. The

present volume is intended to communicate to a wider mathematical audience the

scope and permanence of Harish-Chandra's mathematical legacy. For about three

decades starting from 1950 he created, essentially all by himself, a monumental

structure of representation theory and harmonic analysis associated with reductive

groups and their homogeneous spaces. It is certainly of great interest to retrace his

thinking and to try to understand the architecture of his epoch-making achievement.

Actually for those who are interested in detailed accounts of his life and work there

are now available many sources: my introduction as well as the expositions by

Wallach and Howe in Volume I of his Collected Papers [Hi], my account of his

life, work, and personality [V], the articles of Langlands [LI], [L2], the eulogies

delivered on the occasion of a memorial conference for him in Princeton in 1984

[H2] which are being reprinted here, the recollections of Borel [Bo], and the article

of Herb [He].

Harish-Chandra's ideas, results, and techniques have influenced an entire gen-

eration of mathematicians in a way that has made the subject of representation

theory and harmonic analysis grow into one of the central areas of mathematics

today. However it is not within my capacity to describe all the developments that

2000 Mathematics Subject Classification. Primary 22E46.

Key words and phrases. Unitary representation, orbital integral, limit formula, Harish-

Chandra homomorphism, analytic vectors, subquotient theorem, characters, regularity theorem,

method of descent, discrete series, the Harish-Chandra character formula, cusp forms, constant

term, Plancherel formula.

©2000 American Mathematical Society

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http://dx.doi.org/10.1090/pspum/068/1767887