HARISH-CHANDRA

(b. Kanpur, India 11 October 1923; d. Princeton, New Jersey. 16 October 1983)

mathematics.

Harish-Chandra was one of the most profound mathematicians of his time. Although he started his scientific career as a theoretical physicist, he was always preoccupied with the purely mathematical aspects of physical theories, and eventually worked entirely in mathematics. His work, in algebra and analysis, is concerned almost exclusively with the theory of representations of groups and their harmonic analysis. This area of mathematics is at the interface of such varied disciplines as physics, number theory, and geometry; and some of the greatest mathematicians—such as Georg Frobenius, Élie Cartan, Hermann Weyl, I. M. Gel’fand, Atle Selberg, and Robert Langlands—have been attracted to it. Harish-Chandra’s work is widely regarded as a brilliant and enduring part of it. That representation theory and harmonic analysis are among the most central and active areas of interest in present-day mathematics is partly due to his lifelong efforts dating Form the 1940’s. He had his early scientific training in India but settled in the United States. Where almost all of his work was done. No mathematician of comparable stature has arisen from India in the second half of the twentieth century.

Harish-Chandra was born into an upper-middleclass family. His father, Chandra Kishore, was a civil engineer in the government of Uttar Pradesh (then known as the United Provinces); his mother, Satyagati Seth Chandrarani, was the daughter of a lawyer. Harish-Chandra was a brilliant student, and graduated with an M.Sc. from the university of Allahabad in 1943. Although the educational system under the British colonial administration was mostly sterile, there were some exceptions here and there, mainly due to the efforts of a few remarkable but isolated individuals, such as C.V. Raman in Bangalore and P.C. Mahalanobis in Calcutta. Another was K.S. Krishnan, a physicist at the University of Allahabad. Harish-Chandra studied under Krishnan, whose influence stayed with him throughout his life.

From Allahabad, Harish-Chandra went to Bangalore in southern India as a research student with H.J. Bhabha, a leading theoretical physicist who later created the Tata Institute of Fundamental Research and developed it into a world-class research institution in mathematics and physics. In 1945 Harish-Chandra went to England to work with P.A.M. Dirac at Cambridge. His thesis, on the classification of irreducible representation of the Lorentz group, gave clear indications that he was already becoming fascinated with the purely mathematical. In the year 1947–1948 he was at the Institute for Advanced Study at Princeton as an assistant to Dirac, who was a visiting professor there. In 1950 he went to Columbia University, remaining there until 1963. He then returned to the Institute for Advanced Study as a permanent member, a position he held until his death. In 1968 Harish-Chandra was named the I.B.M.-von Neumann professor of mathematics at the institute. He was elected a fellow of the Royal Society in 1973 and a member of the U.S. National Academy of Sciences in 1981. He was awarded honorary doctorates by Delhi University in 1973 and Yale University in 1981.

In 1952 Harish-Chandra married Lalitha Kale; they had two daughters.

Harish-Chandra’s health was always fragile, and his health problems were compounded by the relentlessly ascetic and overwhelmingly intense nature of his work patterns. He suffered a heart attack in 1969, but continued to work on the central questions, as he saw them, of representation theory. But the damage to his heart was irreversible and his problems grew worse, resulting in a fatal heart seizure as he was walking in the woods near the institute at Princeton.

The origins of the theory of group representations go back at least to the work of Georg Frobenius in the nineteenth century; its main problem is to represent abstractly given groups by concrete matrices in such a way that the group operation corresponds to multiplication of matrices. Originally only finite groups and matrices of finite size were considered. However, with the rise of quantum mechanics it became increasingly clear that infinite groups and matrices have to be included. Indeed, one of the fundamental techniques for understanding very complex interacting systems of highly energetic elementary particles is to study the groups of symmetries that leave the system unchanged, and to use this study to limit the possibilities.

In order to apply this technique, it is indispensable to have a complete knowledge of the representations of the symmetry groups. The groups that are important in these problems, such as the rotation and Lorentz groups and their variants, are examples of what mathematicians call simple groups. The simple groups were classified by Cartan and Wilhelm Killing in the nineteenth century, and by 1925 all their finite dimensional representations were classified by Cartan and Weyl. What Harish-Chandra did was to construct the most fundamental infinite dimensional representations of all the simple groups and use them to carry out the harmonic analysis of functions and generalized functions in these groups. The technical problems that he had to overcome in completing this project involved entirely novel aspects of the theory of differential equations on group manifolds.

High-energy physics is not the only area where representation theory of simple groups is a crucialingredient. Modern number theory is another, and current formulations of the problem of understanding the arithmetic of Galois extensions of algebraic number fields make essential use of the language and results of the representation theory. The classical number theoretic investigations, culminating in the work of David Hilbert. Teiji Takagi, Emil Artin, Helmut Hasse, and others, had obtained a marvelous description of the Abelian extensions of number fields. One of the great achievements of modern number theory is the discovery that similar descriptions of all extensions (Abelian or not) of number fields ultimately depend on establishing certain natural correspondences between representations of the Galois groups and appropriate representations of certain simple groups associated to the number fields.

Although there had been many attempts to study infinite-dimensional representations of groups before Harish-Chandra, it was he who began the systematic theory of representations of all simple groups. Unlike his predecessors, who worked with special groups and ad hoc methods, Harish-Chandra realized immediately that it was essential to develop an approach that integrated the algebraic, geometric, and analytic aspects of simple groups. For instance, his papers were the first to consider representations of nontrivial infinite-dimensional associative algebras. Quite early in his work he made the major discovery that one can associate to any irreducible representation of a simple group a character that is the trace of the matrices that represent the group elements. Characters are not new, of course; both Frobenius and Weyl had worked with them and obtained beautiful formulas for them, but they worked with matrices of finite size and there was no conceptual difficulty in defining the characters and developing their theory. But the Harish-Chandra character is more subtle and singular, being the trace of an infinite matrix; indeed, it is a nontrivial matter even to define it, and the only tools available for its study are the differential equations satisfied by it.

The cornerstone of Harish-Chandra’s work is a profound and astonishingly complete study of the solutions of these differential equations that led him to a complete description of all the fundamental characters. Complicating the development was the fact that the characters are a priori functions only in a very generalized sense, and Harish-Chandra had to prove ex post facto that they are functions in the classical sense. This result, known as the Harish-Chandra regularity theorem, is one of his greatest achievements and its influence pervades the entire subject. In spite of repeated efforts by many people, it has not been possible to deduce this theorem as a consequence of general principles from the theory of differential equations.

In order to carry out the harmonic analysis of functions on the simple groups in terms of the characters, it is necessary to understand how the elementary functions that arise from the characters behave at infinitely distant parts of the group manifold. Thesecond major achievement of HarishChandra was his complete solution to this problem and the explicit harmonic analysis that flowed from this asymptotic theory. Roughly speaking, these elementary functions can be expanded in an infinite series at infinity, and the leading terms of these expansions determine in a very simple and explicit manner the weights with which the various characters enter the harmonic analysis of the delta function on the group.

BIBLIOGRAPHY

I. Original Works. Harish-Chandra’s works were brought together in his Collected Papers, V.S. Varadarajan, ed., 4 vols. (New York and Berlin, 1984).

II. Secondary Literature. Robert P. Langlands, in Biographical Memoirs of Fellows of the Royal Society, 31 (1985), 199–225; and V.S. Varadarajan. “HarishChandra (1923–1983),” The Mathematical Intelligencer, 6, no. 3 (1984), 9–19.

V. S. Varadarajan