Harmonic Analysis and Group Representations, Volume 47, Number 1

26 NOTICES OF THE AMS VOLUME 47, NUMBER 1 H armonic analysis can be interpreted broadly as a general principle that relates geometric objects and spectral objects. The two kinds of objects are sometimes related by explicit formulas, and sometimes simply by parallel theories. This principle runs throughout much of mathematics. The rather impressionistic table at the top of the opposite page provides illustrations from different areas. The table gives me a pretext to say a word about the Langlands program. In very general terms, the Langlands program can be viewed as a series of farreaching but quite precise conjectures, which describe relationships among two kinds of spectral objects—motives and automorphic representations—at the end of the table. Wiles’s spectacular work on the Shimura-Taniyama-Weil conjecture, which established the proof of Fermat’s Last Theorem, can be regarded as confirmation of such a relationship in the case of elliptic curves. In general, the arithmetic information wrapped up in motives comes from solutions of polynomial equations with rational coefficients. It would not seem to be amenable to any sort of classification. The analytic information from automorphic representations, on the other hand, is backed up by the rigid structure of Lie theory. The Langlands program represents a profound organizing scheme for fundamental arithmetic data in terms of highly structured analytic data. I am going to devote most of this article to a short introduction to the work of HarishChandra. I have been motivated by the following three considerations. (i) Harish-Chandra’s monumental contributions to representation theory are the analytic foundation of the Langlands program. For many people, they are the most serious obstacle to being able to work on the many problems that arise from Langlands’s conjectures. (ii) The view of harmonic analysis introduced above, at least insofar as it pertains to group representations, was a cornerstone of HarishChandra’s philosophy. (iii) It is more than fifteen years since the death of Harish-Chandra. As the creation of one of the great mathematicians of our time, his work deserves to be much better known. I shall spend most of the article discussing Harish-Chandra’s ultimate solution of what he long regarded as the central problem of representation theory, the Plancherel formula for real groups. I shall then return briefly to the Langlands program, where I shall try to give a sense of the role played by Harish-Chandra’s work.