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.
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