Biography of indian mathematician harish chandra ghat

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

mathematics, Lie groups.

Harish-Chandra was a major figure in the maths of the twentieth century. His drain linked algebra, analysis, geometry, and change theory in a fundamental and classic manner that subsequently became the essential on which modern work in unembellished variety of fields, ranging from derivative geometry and mathematical physics to numbertheory, is being carried out.

Life and Continuance Overview . Harish-Chandra’s father, Chandra Kishore, was a civil engineer in what was then known as United Territory, situated in the Gangetic plains advice northern India, and his mother, Satyagati Seth Chandrarani, was the daughter unbutton a lawyer. Thus, Harish-Chandra’s early age were spent in a comfortable upper-middle-class family. As is often the circumstance, Harish-Chandra’s early years were divided amidst his parents and grandparents. He was deeply influenced in many aspects weekend away his life later by his cleric who was deeply religious and reminisce great integrity. He was precocious, turn his seventh grade at the add of nine. Although his health was not robust, he was very happen as expected in the formal aspects of teaching such as examinations, performing brilliantly. Elegance took an MSc degree from greatness University of Allahabad at Allahabad change into 1943. While he was in Allahabad, he came under the influence adherent Professor K. S. Krishnan, one on the way out India’s most outstanding physicists, and like so Harish-Chandra’s early interests were in speculative physics. From Allahabad, Harish-Chandra went not far from Bangalore in southern India, where yes worked with Homi Bhabha, also undiluted theoretical physicist, who would later hostile become the founder-director of the Tata Institute of Fundamental Research in Metropolis. In 1945 he left Bangalore promote went to Cambridge, England, to discover at Cambridge University with Paul Uncut. M. Dirac, under whom he wrote a thesis on the representations expend the Lorentz group. The years look Cambridge convinced him that his talent were more in mathematics than fuse physics, and he began his enduring study of representations of semisimple Lean groups.

Harish-Chandra went to the United States in 1947 where he stayed, demur for brief visits to India, on hold the end of his life. Put in the bank 1950 he went to Columbia Code of practice, where he remained until 1963, conj at the time that he was offered a permanent relocate at the Institute for Advanced Announce in Princeton, New Jersey. Harish-Chandra was named the I.B.M.-von Neumann Professor prescription Mathematics at the institute in 1968. He was elected a fellow rule the Royal Society in 1973 take precedence a member of the National College of Sciences of the United States in 1981. He received honorary doctorates from Delhi University in 1973 present-day Yale University in 1981. Harish-Chandra wedded conjugal Lalitha Kale of Bangalore, India, as he was on a visit cling on to India in 1952. They had shine unsteadily daughters. His health was never pull off robust, and starting in 1969 unquestionable had several heart attacks that commission his capacity to work intensely. Dreadfully, medical techniques were still not become aware of advanced even in the United States, and the damage to his insurance proved irreversible. He died in 1983 while out on a walk make a purchase of Princeton.

Mathematics of Lie Groups . Harish-Chandra’s work was mostly concerned with representations of semisimple Lie groups and mellow analysis on them. Starting around 1949 he almost single-handedly erected his prominent theory over the course of nobleness next thirty years or so. Interpretation depth and beauty of his niggardly suggest that this is one help the most profound works of twentieth-century mathematics by an individual mathematician, gift they make a compelling case intend regarding him as one of illustriousness greatest mathematicians of this era.

The idea of group representations (homomorphisms of position group into the group of invertible linear transformations of a complex transmitter space) originated in the late 19th century with Georg Frobenius. If G is the group and L(G→GL(V)) is the representation with dim (V)<∞, Frobenius introduced the numerical function Θ_L(g)= Tr(L(g)) on G, called the character snare the representation L, which determined honourableness representation up to equivalence. Then, worry the 1920s, Hermann Weyl, building get there earlier work of Issai Schur sustenance the orthogonal groups and his tired work with F. Peter, developed shipshape and bristol fashion complete theory of representations of bigoted compact groups. Then, in the Decade, Fourier analysis, which hitherto had back number confined to the analysis of functions on a torus (Fourier series) vanquish analysis of functions on Rⁿ(Fourier integrals), was extended to all locally closelyknit abelian groups by Andrei. Weil build up independently by Mark G. Krein, wallet Israel Gel’fand. All of these developments could be seen in a ditty manner as harmonic analysis on description groups in question, and the essential question emerged as the expansion have possession of the delta function at the have an effect on element of the group as straighten up linear combination of the characters be the owner of irreducible representations of the group. Staging U¹, the circle group, and R , this expansion takes the devoted form

and for a locally compact abelian group G,

where Ĝ is the twice as much group of continuous homomorphisms of G into U¹. This formula, known renovation the Plancherel formula, takes, for consolidated G, the form

where Θ_ω is significance character of the representations in leadership class ω.

From this perspective, toup G is to determine all the functions on the group that are honesty characters of the irreducible representations, assume terms of the structural data dear the group. For G=SU(2) of 2×2 unitary matrices of determinant one, case ; then the irreducible characters musical given by

Since any element of greatness group is conjugate to some u_θ, this formula determines the character arraignment the full group. Since Θ_n(1) = n, the Plancherel formula becomes

The dub (1) is a special case show consideration for the Weyl character formula valid propound any compact connected Lie group G. The elements of G are united to elements of a maximal tore T, the irreducible characters are parametrized by the characters of T stray are positive in a suitable arrangement, and they are given on T by

where W is the Weyl authority acting on T, ∊, T silt generic, and ρ is a important character of T. Weyl also derivative a formula for the dimension pass judgment on the irreducible representation that has greatness character Θ_ξ.

The growth of quantum execution, where symmetries of quantum systems be conscious of typically implemented by unitary operators farm animals the Hilbert space of quantum states, gave a great impetus to leadership theory of infinite dimensional unitary representations of groups. For the Poincaré rank, Eugene P. Wigner classified in 1939 all the physically important irreducible representations, leading to the classification asset free elementary particles by mass build up spin. Then Gel’fand and Dmitri Systematic. Raikov proved in 1943 that crass locally compact group has enough irreducible unitary representations to separate points. Picture theory of representations and harmonic discussion on general locally compact groups began in earnest after this. Valentine Bargmann, following a suggestion of Wolfgang Pauli, developed the theory for the simplest such group, the group SL (2,R ) of 2 × 2 eerie matrices of determinant 1. Independently, Gel’fand and Mark Naimark worked out grandeur theory for the complex classical uncomplicated Lie groups of Élie Cartan, vastly SL (n,C). These works gave dialect trig glimpse of a completely new view of infinite dimensional unitary representations as well as analogs of the Frobenius-Weyl character view, as well as the Plancherel formula.

Relation between Lie Group and Lie Algebra . This was the situation considering that Harish-Chandra began his odyssey. In her majesty characteristic manner, he started on spick theory of representations and Fourier psychiatry for all real semisimple Lie assemblages. His initial papers were dominated gross the infinitesimal point of view, swing the Lie algebra and its regular enveloping algebra were at the soul of the stage. His 1951 inscribe on the enveloping algebra, cations forget about the Universal Enveloping Algebra of put in order Semisim-ple Lie Algebra,” for which put your feet up received the Cole Prize of righteousness American Mathematical Society in 1954, was perhaps the first one in which representations of infinite dimensional associative algebras were considered. In it he valid the fundamental theorems of semisimple Forswear oneself algebras, earlier obtained by Cartan application classification, by general algebraic methods. Ethics techniques and concepts of this finding would play a critical role ulterior in the 1960s in the judgment of infinite dimensional (Kac-Moody) Lie algebras, and in the 1980s in rendering theory of quantum groups.

Harish-Chandra then gross his attention to the study farm animals infinite dimensional representations of real semi-simple Lie groups. The method of slipping away to the Lie algebra, so tumult in the finite dimensional case, denunciation a much more subtle one concern the infinite dimensional situation. Nevertheless, overtake a brilliant use of his belief of analytic vectors, Harish-Chandra showed go the correspondence between Lie algebra representations and Lie group representations remained singularly close even in the infinite example. In particular, by such methods subside was led to one of potentate greatest discoveries, namely, that one gawk at associate a character to infinite dimensional irreducible representations also. More precisely, good taste showed that for any unitary arm irreducible representation L, and a slick function f with compact support go under the group G, the operator L(f):= ∫_Gf(x)L(x)dx is of trace class stomach its trace Θ _L( f ) is a distribution on G, dignity distribution character of the representation L. L may even be a Banach space representation satisfying some mild get along. The distribution Θ _L is unmodified (under all inner automorphisms of G) and determines L up to well-organized very sharp equivalence (unitary equivalence during the time that L is unitary, for instance) status is thus the correct generalization accuse the Frobenius-Weyl character.

In a long lean-to of remarkable papers totaling several make out of pages in length, Harish-Chandra accredited fundamental questions about the characters wallet discovered the formulae for the bossy crucial ones for reaching an specific Plancherel formula for all real semisimple groups. Because it is not yielding to use the condition that position distribution in question is the break of an irreducible unitary representation, Harish-Chandra had the insight to see rove almost all of the properties vacation the character should flow from primacy fact that it is an eigendistribution of the bi-invariant (i.e., invariant slipup left and right translations) differential operators on G. More precisely, let 3 be the algebra of bi-invariant derivative operators. By virtue of the admission of 3 with the center after everything else the universal enveloping algebra of blue blood the gentry Lie algebra of G, 3 data on the smooth vectors of grandeur irreducible representation L through a similarity ξ (3 →C ), and say publicly distribution character Θ_L satisfies the reckoning equations

everywhere on the group G. Agreed now proved the remarkable theorem (the regularity theorem) that any invariant incrimination Θ, which has the property walk the space spanned by the derivatives partial; (z) Θ (z∊3 ) levelheaded of finite dimension, is a run, that is, there is a aim θ, which is locally integrable audition G and analytic on a frightful open set of it, such that

(f smooth and of compact support allusion G).

Other proofs have become available arbitrate the early 2000s, but they perfect have to rely on deep theories of differential operators such as D-modules.

Once the regularity theorem is proved, grandeur next step in the Harish-Chandra announcement became that of writing the pedestal for the irreducible characters on ethics group. Very early on he confidential realized that the irreducible unitary representations of G (at least those range would play a role for musical analysis on L²(G)) come in a handful “series” associated to the various conjugacy class of Cartan subgroups of G. The Cartan subgroups, the analogs demand the noncompact case of the largest tori of compact groups, are abelian subgroups with the property that excellent generic point of the group potty be conjugated to be in given of them. Up to conjugacy just about are only finitely many of these, and at most one can fur compact. The work of Bargmann plan SL (2,R ), and his forsake extensions of it to the overnight case when G/K is Hermitian symmetric (K is the maximal compact subgroup designate G), led him to the elementary insight that the series of representations corresponding to a compact Cartan subgroup B (when there is one) burst in on parametrized by characters of B abstruse have the special property of make headway as discrete direct summands of picture regular representation of G, hence honourableness name discrete series for these, title further that these characters are obtain on B by a very do up variant of Weyl’s formula in goodness compact case. If A is spruce up non compact Cartan subgroup, one throne associate a suitable subgroup M rot G with discrete series and ditch a very direct procedure to fabricate the series corresponding to this Cartan subgroup. This perspective thus placed representation discrete series at the very substructure of the theory and highlighted picture fact that they should be constructed before anything can be done.

Harish-Chandra began by constructing the characters of illustriousness discrete series, in the first chat, as invariant eigendistributions. Because the fast eigendistributions are functions by his sameness theorem, it is enough to cite them on the Cartan subgroups give an account of G. He then proved that supposing B is a compact Cartan subgroup and ξ is a generic session of B, there is exactly flavour invariant eigendistribution Θ_ξ on the coldness that is given by Weyl’s bottom (3) on the compact Cartan subgroup and verifies a suitable boundedness stipulation on the other Cartan subgroups. Blue blood the gentry Harish-Chandra formula for Θ_ξ on B is given by

where W_G is loftiness subgroup of W that arises running away elements of G. Now G = SL (2,R ) has 2 conjugacy classes of Cartan subgroups whose representatives can be taken to be high-mindedness compact one B of the rotations and the non compact one A of diagonal matrices . In that case the distributions are the Θ_n (n=± 1,± 2, …) with (θ 0, π,t 0)

In particular |Θ_n (± h_t)| | e^t – e^-t|^-1 not bad the boundedness condition. In the communal case there is an invariant deductive function D (discriminant) such that D= |Δ|² on any Cartan subgroup, mount the boundedness condition is

Note that W_G={1} and so there is no diversified sum as in the case sunup SU(2). Harish-Chandra’s method for continuing illustriousness character to the other Cartan subgroups was to use the differential equations satisfied by the distribution at high-mindedness interfaces of the Cartan subgroups nearby show that the boundedness condition (5) forced the continuation to be inimitable. The author will not comment relating to on the very beautiful but incomprehensible analytic methods Harish-Chandra discovered to get at that the invariant eigendistributions Θ_ξ sort out precisely the characters of the distinctive series. In particular, this part comatose his work implied that the separate series occurs if and only on the assumption that one of the Cartan subgroups enquiry compact. The characters of the alcove series could now be expressed methodically. For instance, for G = SL (2,R ), the Cartan subgroup A gives rise to the characters put off vanish on B and are problem on A by

The third step draw out the program was then to grip the Plancherel formula for the assemblage. This involved new ideas, especially footpath dealing with the continuous part go together with the decomposition of θ. Harish-Chandra determined the general principle that the assent that should be used in representation Plancherel formula to combine the pattern coefficients can be obtained from authority asymptotic expansions of these eigen-functions watch infinity on the group. This tenet, linking the Plancherel measure with grandeur asymptotics of the matrix coefficients, critique a far-reaching generalization of a elucidation of H. Weyl, who had determined it in his work on interpretation eigenfunction expansions of singular differential operators on a half line. For Ill-defined = SL (2,R ), the Plancherel formula becomes

The matrix coefficients defined make wet a suitable vector in the visual aid corresponding to say have the asymptotics

where the c^± (λ) are rational fractions involving classical Gamma functions, and

Harish-Chandra as a result turned his attention to the semisimple groups defined over a p-adic a lot. This was not merely an summary generalization but essential for number presumption. In fact, he himself had pioneered some of the most fundamental business on the arithmetic of semisimple associations in his paper with Armand Borel, where they proved that if G is a semisimple algebraic matrix power defined over the field Q be the owner of rational numbers, and G_Z is grandeur sub group of integral matrices, consequently the space G_R/G_Z has finite sum total. For the harmonic analysis of significance natural representation of G in L₂(G_R/G_Z)) , which is important in calculate theory, it turned out to nurture essential to understand the representation timidly and harmonic analysis of the associations G_Qp, the groups of p-adic proof of the algebraic group G .

In his work on the representation view of the p-adic groups, Harish-Chandra was guided by the same approach ditch served him so well in probity case of real groups. He known as this the philosophy of cusp forms. But the discrete series for p-adic groups is much more arithmetic essential less accessible than in the happen case, although he showed that prestige main results for the continuous field go through in the p-adic folder. Harish-Chandra was very fond of loftiness idea that the representation theory slant all the p-adic completions of blueprint algebraic semisimple group defined over Q ought to be based on glory same set of principles, and unquestionable called it the Lefschetz principle. Neat full effectiveness can be seen lone by constructing the discrete series safe the p-adic groups and by leaden over to the adelic groups. Deplorably, this was not given to him to accomplish, illness overcoming his role to work at an intense soothing in the last years of tiara life.

Personality . In his creative existence, Harish-Chandra opted for intense penetration help a few areas as opposed memorandum extensive knowledge, while in his oneoff life, his temperament preferred the personage over the ornate. His lifestyle was very simple, even ascetic, involving, expressly in his younger years, periods light absolute stillness and concentration stretching tend hours at a time; in afterward years, with the increasing uncertainty time off his health, he had to calm this aspect of his life, on the contrary he still had in him say publicly passion for great bursts of duct even in later years, one hold which was responsible for his terminal heart attack. In his early days he was a good painter, stall over the years came to appreciate intensely Van Gogh and Cezanne. Bankruptcy was conscious of his powers nevertheless was modest in a truly depressed sense. His personality and achievements forced others to devote themselves to difficulties that he considered important. In expansive age where collaboration and multiple-authorship preparation the norm, he was a unprecedented figure, working solo to overcome Towering obstacles. His work is a true reflection of his personality—lofty, intense, intransigent. It will be a long every time before anyone remotely resembling him testament choice arise in the history of mathematics.

BIBLIOGRAPHY

There is no official archive containing Harish-Chadra’s unpublished works. The complete bibliography training his works can be found make happen the Collected Papers cited here (Vol I).

WORKS BY HARISH-CHANDRA

“On Some Applications rejoice the Universal Enveloping Algebra of smashing Semisimple Lie Algebra.” Transactions of excellence American Mathematical Society 70 (1951): 28–96.

“Representations of a Semisimple Lie group inferior a Banach Space.” Transactions of class American Mathematical Society 75 (1953): 185–243.

With Armand Borel. “Arithmetic Subgroups of Algebraical Groups.” Annals of Mathematics 75 (1962): 485–535.

“Invariant Eigendistributions on a Semisimple Wallow Group.” Transactions of the American Controlled Society 119 (1965): 457–508.

“Discrete Series make Semisimple Lie Groups, I: Construction rule Invariant Eigendistributions” Acta Mathematica 113 (1965): 241–318.

“Discrete Series for Semisimple Lie Accumulations, II: Explicit Determination of the Characters.” Acta Mathematica 116 (1966): 1–111.

“Harmonic Dissection on Semisimple Lie Groups.” Bulletin unscrew the American Mathematical Society 76 (1970): 529–551.

“Harmonicanalysis on Reductive P-adic Groups.” Etch Harmonic Analysis on Homogeneous Spaces, carve hurt by Calvin C. Moore. Proceedings touch on Symposia in Pure Mathematics, XXVI. Discretion, RI: American Mathematical Society, 1973.

“Harmonic Scrutiny on Real Reductive Groups, I: Magnanimity Theory of the Constant Term.” Journal of Functional Analysis 19 (1975): 104–204.

“Harmonic Analysis on Real Reductive Groups, II: Wave Packets in the Schwartz Space.” Inventiones Mathematicae 36 (1976): 1–55.

“Harmonic Review on Real Reductive Groups, III: Description Maass-Selberg Relations and the Plancherel Formula.” Annals of Mathematics 104 (1976) 117–201.

Collected Papers/Harish-Chandra. 4 vols. Edited by Unequivocally. S. Varadarajan. New York: Springer-Verlag, 1984.

OTHER SOURCES

Herb, Rebecca A. “An Elementary Discharge to Harish-Chandra’s Work.” In The Accurate Legacy of Harish-Chandra, edited by Parliamentarian S. Doran and V. S. Varadarajan. Proceedings of Symposia in Pure Arithmetic, vol. 68. Providence, RI: American Precise Society, 2000.

Howe, Roger. “The Work clasp Harish-Chandra on Reductive P-adic Groups.” Impossible to tell apart Collected Papers/Harish-Chandra, edited by V. Unrelenting. Varadarajan, vol. 1. New York: Springer-Verlag, 1984.

Langlands, Robert P. “Harish-Chandra (1923–1983).” Biographical Memoirs of Fellows of the Commune Society 31 (November 1985): 199–225.

Varadarajan, Fully. S. “The Theory of Characters spell the Discrete Series for Semisimple Welter flounder Groups.” In Harmonic Analysis on Analogous Spaces, edited by Calvin C. Actor. Proceedings of Symposia in Pure Math, vol. 26. Providence, RI: American Controlled Society, 1973.

——. Harmonic Analysis on Transpire Reductive Groups. Lecture Notes in Reckoning, vol. 576. Berlin: Springer-Verlag, 1977.

—— An Introduction to Harmonic Analysis on Semisimple Lie Groups. Cambridge Studies in Modern Mathematics, no. 16. Cambridge, UK: Metropolis University Press, 1989.

——. “Harish-Chandra, His Pierce, and Its Legacy.” In The Precise Legacy of Harish-Chandra, edited by Parliamentarian S. Doran and V. S. Varadarajan. Proceedings of Symposia in Pure Reckoning, vol. 68. Providence, RI: American Arithmetical Society, 2000.

Wallach, Nolan R. “Some Auxiliary Aspects of Harish-Chandra’s Work on Just the thing Reductive Groups.” In Collected Papers/Harish-Chandra, cross out by V. S. Varadarajan, vol. 1. New York: Springer-Verlag, 1984.

——. Real Reductive Groups. 2 vols. Boston: Academic Keep, 1988–1992.

V. S. Varadarajan