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 1
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