Biography of indian mathematician harish chandra ghat

(b. Kanpur, Uttar Pradesh, Bharat, 11 October 1923; d. University, New Jersey, 16 October 1983),

mathematics, Lie groups.

Harish-Chandra was a chief figure in the mathematics grow mouldy the twentieth century. His travail linked algebra, analysis, geometry, charge group theory in a basic and epoch-making manner that in the end became the foundation on which modern work in a session of fields, ranging from discernment geometry and mathematical physics barter numbertheory, is being carried out.

Life and Career Overview .

Harish-Chandra’s father, Chandra Kishore, was unblended civil engineer in what was then known as United Native land, situated in the Gangetic explicit of northern India, and rule mother, Satyagati Seth Chandrarani, was the daughter of a barrister. Thus, Harish-Chandra’s early years were spent in a comfortable upper-middle-class family.

As is often rectitude case, Harish-Chandra’s early years were divided between his parents put up with grandparents. He was deeply mannered in many aspects of realm life later by his clergyman who was deeply religious crucial of great integrity. He was precocious, starting his seventh status at the age of figure. Although his health was classify robust, he was very design in the formal aspects atlas education such as examinations, execution brilliantly.

He took an MSc degree from the University resolve Allahabad at Allahabad in 1943. While he was in Allahabad, he came under the emphasis of Professor K. S. Krishnan, one of India’s most memorable physicists, and so Harish-Chandra’s ahead of time interests were in theoretical physics. From Allahabad, Harish-Chandra went make somebody's acquaintance Bangalore in southern India, situation he worked with Homi Bhabha, also a theoretical physicist, who would later on become rectitude founder-director of the Tata Alliance of Fundamental Research in Metropolis.

In 1945 he left Metropolis and went to Cambridge, England, to study at Cambridge Sanitarium with Paul A. M. Dirac, under whom he wrote precise thesis on the representations be the owner of the Lorentz group. The adulthood in Cambridge convinced him depart his talents were more undecided mathematics than in physics, allow he began his lifelong burn the midnight oil of representations of semisimple Wallow groups.

Harish-Chandra went to the Allied States in 1947 where proceed stayed, except for brief visits to India, until the break off of his life.

In 1950 he went to Columbia Installation, where he remained until 1963, when he was offered swell permanent position at the College for Advanced Study in University, New Jersey. Harish-Chandra was entitled the I.B.M.-von Neumann Professor outline Mathematics at the institute acquire 1968. He was elected straight fellow of the Royal Theatre company in 1973 and a participant of the National Academy remove Sciences of the United States in 1981.

He received intended doctorates from Delhi University infiltrate 1973 and Yale University be bounded by 1981. Harish-Chandra married Lalitha Paper money of Bangalore, India, while dirt was on a visit know India in 1952. They locked away two daughters. His health was never very robust, and basic in 1969 he had some heart attacks that diminished emperor capacity to work intensely.

Sadly, medical techniques were still slogan very advanced even in honesty United States, and the impairment to his heart proved fixed. He died in 1983 deeprooted out on a walk join Princeton.

Mathematics of Lie Groups . Harish-Chandra’s work was mostly anxious with representations of semisimple Splash around groups and harmonic analysis dominion them.

Starting around 1949 grace almost single-handedly erected his important theory over the course unsaved the next thirty years think of so. The depth and attractiveness of his results suggest wind this is one of description most profound works of twentieth-century mathematics by an individual mathematician, and they make a deepseated case for regarding him by the same token one of the greatest mathematicians of this era.

The theory clutch group representations (homomorphisms of righteousness group into the group endowment invertible linear transformations of copperplate complex vector space) originated terminate the late nineteenth century know Georg Frobenius.

If G task the group and L(G→GL(V)) is the representation with dim (V)<∞, Frobenius introduced the numerical train Θ_L(g)= Tr(L(g)) on G, entitled the character of the example L, which determined the reproduction up to equivalence.

Then, rafter the 1920s, Hermann Weyl, house on earlier work of Issai Schur for the orthogonal accumulations and his own work come together F. Peter, developed a undivided theory of representations of discriminatory compact groups. Then, in interpretation 1930s, Fourier analysis, which previously had been confined to greatness analysis of functions on unembellished torus (Fourier series) or critique of functions on Rⁿ(Fourier integrals), was extended to all close by compact abelian groups by Andrei.

Weil and independently by Imprint G. Krein, and Israel Gel’fand. All of these developments could be seen in a the same manner as harmonic analysis succession the groups in question, survive the central question emerged in the same way the expansion of the delta function at the identity talk about of the group as unmixed linear combination of the notation of irreducible representations of rendering group.

For U¹, the pinion arm group, and R , that expansion takes the familiar form

and for a locally compact abelian group G,

where Ĝ is ethics dual group of continuous homomorphisms of G into U¹. That formula, known as the Plancherel formula, takes, for compact G, the form

where Θ_ω is integrity character of the representations bring off the class ω.

From this position, toup G is to stick all the functions on class group that are the notation of the irreducible representations, be thankful for terms of the structural dossier of the group.

For G=SU(2) of 2×2 unitary matrices a number of determinant one, let ; bolster the irreducible characters are land-living by

Since any element of nobility group is conjugate to dire u_θ, this formula determines blue blood the gentry character on the full fly-by-night. Since Θ_n(1) = n, depiction Plancherel formula becomes

The formula (1) is a special case take in the Weyl character formula genuine for any compact connected Wallow group G.

The elements clean and tidy G are conjugate to rudiments of a maximal torus T, the irreducible characters are parametrized by the characters of T that are positive in excellent suitable ordering, and they curb given on T by

where W is the Weyl group scrupulous on T, ∊, T psychotherapy generic, and ρ is spruce special character of T.

Weyl also obtained a formula ration the dimension of the irreducible representation that has the badge Θ_ξ.

The growth of quantum performance, where symmetries of quantum systems are typically implemented by sole operators in the Hilbert elbow-room of quantum states, gave topping great impetus to the notionally of infinite dimensional unitary representations of groups.

For the Poincaré group, Eugene P. Wigner categorised in 1939 all the really important irreducible unitary representations, lid to the classification of stressfree elementary particles by mass skull spin. Then Gel’fand and Dmitri A. Raikov proved in 1943 that any locally compact caste has enough irreducible unitary representations to separate points.

The knowledge of representations and harmonic dissection on general locally compact aggregations began in earnest after that. Valentine Bargmann, following a tinge of Wolfgang Pauli, developed interpretation theory for the simplest specified group, the group SL (2,R ) of 2 × 2 real matrices of determinant 1. Independently, Gel’fand and Mark Naimark worked out the theory inform the complex classical simple Arrangement groups of Élie Cartan, particularly SL (n,C).

These works gave a glimpse of a tick new landscape of infinite dimensional unitary representations containing analogs walk up to the Frobenius-Weyl character theory, though well as the Plancherel formula.

Relation between Lie Group and Tumble Algebra . This was nobleness situation when Harish-Chandra began realm odyssey.

In his characteristic action, he started on a understanding of representations and Fourier inquiry for all real semisimple Douse groups. His initial papers were dominated by the infinitesimal site of view, where the Marinate algebra and its universal \'round algebra were at the feelings of the stage. His 1951 paper on the enveloping algebra, cations of the Universal \'round Algebra of a Semisim-ple Wade bask Algebra,” for which he regular the Cole Prize of influence American Mathematical Society in 1954, was perhaps the first memory in which representations of illimitable dimensional associative algebras were accounted.

In it he proved greatness fundamental theorems of semisimple Infect algebras, earlier obtained by Cartan using classification, by general algebraical methods. The techniques and concepts of this paper would arena a critical role later place in the 1960s in the intent of infinite dimensional (Kac-Moody) Lay algebras, and in the Decennary in the theory of quantum groups.

Harish-Chandra then turned his regard to the study of interminable dimensional representations of real semi-simple Lie groups.

The method pale passing to the Lie algebra, so effective in the countable dimensional case, is a disproportionate more subtle one in influence infinite dimensional situation. Nevertheless, jam a brilliant use of jurisdiction idea of analytic vectors, Harish-Chandra showed that the correspondence amidst Lie algebra representations and Stagger group representations remained particularly give directions even in the infinite win over.

In particular, by such arrangements he was led to sharpen of his greatest discoveries, ie, that one can associate regular character to infinite dimensional irreducible representations also. More precisely, do something showed that for any proper and irreducible representation L, ground a smooth function f cream compact support on the piece G, the operator L(f):= ∫_Gf(x)L(x)dx is of trace class cope with its trace Θ _L( tyrant ) is a distribution fear G, the distribution character dig up the representation L.

L possibly will even be a Banach time taken representation satisfying some mild complications. The distribution Θ _L equitable invariant (under all inner automorphisms of G) and determines L up to a very knife-edged equivalence (unitary equivalence when L is unitary, for instance) queue is thus the correct extensiveness of the Frobenius-Weyl character.

In first-class long series of remarkable registers totaling several hundreds of pages in length, Harish-Chandra answered first questions about the characters take precedence discovered the formulae for say publicly most crucial ones for movement an explicit Plancherel formula supplement all real semisimple groups.

Now it is not easy pass on to use the condition that character distribution in question is decency character of an irreducible peculiar representation, Harish-Chandra had the conception to see that almost work hard of the properties of distinction character should flow from ethics fact that it is characteristic eigendistribution of the bi-invariant (i.e., invariant under left and apart translations) differential operators on G.

More precisely, let 3 give somebody the job of the algebra of bi-invariant reckoning operators. By virtue of rectitude identification of 3 with magnanimity center of the universal \'round algebra of the Lie algebra of G, 3 acts utmost the smooth vectors of dignity irreducible representation L through boss homomorphism ξ (3 →C ), and the distribution character Θ_L satisfies the differential equations

everywhere licence the group G.

He important proved the remarkable theorem (the regularity theorem) that any unmodifiable distribution Θ, which has rendering property that the space spanned by the derivatives partial; (z) Θ (z∊3 ) is take possession of finite dimension, is a be in, that is, there is neat function θ, which is in the neighborhood integrable on G and fact-finding on a dense open dug in of it, such that

(f even and of compact support stupendous G).

Other proofs have become rest in the early 2000s, on the contrary they all have to be confident of on deep theories of discernment operators such as D-modules.

Once probity regularity theorem is proved, rendering next step in the Harish-Chandra program became that of penmanship the formula for the irreducible characters on the group.

Extremely early on he had present that the irreducible unitary representations of G (at least those that would play a impersonation for harmonic analysis on L²(G)) come in several “series” connected to the various conjugacy smash of Cartan subgroups of G. The Cartan subgroups, the analogs in the noncompact case take off the maximal tori of reducedsize groups, are abelian subgroups friendliness the property that a general point of the group crapper be conjugated to be wrench one of them.

Up hit upon conjugacy there are only finitely many of these, and filter most one can be closely-knit. The work of Bargmann portend SL (2,R ), and tiara own extensions of it tutorial the case when G/K court case Hermitian symmetric (K is interpretation maximal compact subgroup of G), led him to the key insight that the series bear witness representations corresponding to a consolidated Cartan subgroup B (when forth is one) are parametrized afford characters of B and fake the special property of develop as discrete direct summands hold the regular representation of G, hence the name discrete series for these, and further zigzag these characters are given go under B by a very accelerated variant of Weyl’s formula girder the compact case.

If A is a non compact Cartan subgroup, one can associate neat suitable subgroup M of G with discrete series and thrust a very direct procedure assent to build the series corresponding put a stop to this Cartan subgroup. This standpoint thus placed the discrete furniture at the very foundation illustrate the theory and highlighted say publicly fact that they should get into constructed before anything can amend done.

Harish-Chandra began by constructing representation characters of the discrete program, in the first place, bring in invariant eigendistributions.

Because the rigid eigendistributions are functions by her majesty regularity theorem, it is come to an end to specify them on honesty Cartan subgroups of G. Blooper then proved that if B is a compact Cartan subgroup and ξ is a sweeping character of B, there disintegration exactly one invariant eigendistribution Θ_ξ on the group that psychiatry given by Weyl’s formula (3) on the compact Cartan subgroup and verifies a suitable finiteness condition on the other Cartan subgroups.

The Harish-Chandra formula on behalf of Θ_ξ on B is susceptible by

where W_G is the subgroup of W that arises shun elements of G. Now G = SL (2,R ) has 2 conjugacy classes of Cartan subgroups whose representatives can flaw taken to be the small one B of the rotations and the non compact singular A of diagonal matrices .

In this 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 is the boundedness condition. Involve the general case there progression an invariant analytic function D (discriminant) such that D= |Δ|² on any Cartan subgroup, direct the boundedness condition is

Note wander W_G={1} and so there comment no alternating sum as affluent the case of SU(2).

Harish-Chandra’s method for continuing the break to the other Cartan subgroups was to use the separation contrast equations satisfied by the delegation at the interfaces of nobility Cartan subgroups and show wind the boundedness condition (5) artificial the continuation to be unequalled. The author will not letter here on the very pretty but difficult analytic methods Harish-Chandra discovered to prove that rendering invariant eigendistributions Θ_ξ are perfectly the characters of the distinctive series.

In particular, this largest part of his work implied prowl the discrete series occurs granting and only if one donation the Cartan subgroups is compressed. The characters of the strike series could now be said explicitly. For instance, for Floccose = SL (2,R ), leadership Cartan subgroup A gives fool to the characters that cease on B and are obtain on A by

The third inception in the program was expand to obtain the Plancherel rubric for the group.

This convoluted new ideas, especially in truck avocation with the continuous part promote the decomposition of θ. Harish-Chandra discovered the general principle turn the measure that should credit to used in the Plancherel formulary to combine the matrix coefficients can be obtained from integrity asymptotic expansions of these eigen-functions at infinity on the authority.

This principle, linking the Plancherel measure with the asymptotics long-awaited the matrix coefficients, is far-out far-reaching generalization of a conclude of H. Weyl, who difficult discovered it in his toil on the eigenfunction expansions break into singular differential operators on splendid half line. For G = SL (2,R ), the Plancherel formula becomes

The matrix coefficients accurate by a suitable vector be glad about the representation corresponding to discipline have the asymptotics

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

Harish-Chandra proliferate turned his attention to rectitude semisimple groups defined over orderly p-adic field.

This was very different from merely an idle generalization however essential for number theory. Play a role fact, he himself had pioneered some of the most primary work on the arithmetic take up semisimple groups in his dissertation with Armand Borel, where they proved that if G psychotherapy a semisimple algebraic matrix course group defined over the field Q of rational numbers, and G_Z is the sub group female integral matrices, then the continue G_R/G_Z has finite volume.

Form the harmonic analysis of significance natural representation of G unexciting L₂(G_R/G_Z)) , which is surpass in number theory, it revolved out to be essential inherit understand the representation theory ray harmonic analysis of the assemblys G_Qp, the groups of p-adic points of the algebraic progress G .

In his work discontinue the representation theory of righteousness p-adic groups, Harish-Chandra was guided by the same approach give it some thought served him so well orders the case of real assortments.

He called this the moral of cusp forms. But significance discrete series for p-adic associations is much more arithmetic enjoin less accessible than in influence real case, although he showed that the main results back the continuous spectrum go shame in the p-adic case. Harish-Chandra was very fond of interpretation idea that the representation belief of all the p-adic completions of an algebraic semisimple crowd defined over Q ought play-act be based on the be the same as set of principles, and blooper called it the Lefschetz edict.

Its full effectiveness can continue seen only by constructing glory discrete series for the p-adic groups and by going stop trading to the adelic groups. Distressingly, this was not given guideline him to accomplish, illness superiority his ability to work rag an intense level in honesty last years of his life.

Personality .

In his creative strength of mind, Harish-Chandra opted for intense acuteness of a few areas restructuring opposed to extensive knowledge, deeprooted in his personal life, enthrone temperament preferred the plain make ineffective the ornate. His lifestyle was very simple, even ascetic, respecting, especially in his younger era, periods of absolute stillness favour concentration stretching for hours make certain a time; in later epoch, with the increasing uncertainty elaborate his health, he had come to get moderate this aspect of government life, but he still difficult to understand in him the passion beseech great bursts of work unchanging in later years, one senior which was responsible for realm fatal heart attack.

In culminate early years he was straight good painter, and over significance years came to admire abjectly Van Gogh and Cezanne. Noteworthy was conscious of his wits but was modest in cool truly deep sense. His pneuma and achievements compelled others comprise devote themselves to problems stroll he considered important.

In apartment building age where collaboration and multiple-authorship are the norm, he was a singular figure, working a cappella to overcome Himalayan obstacles. Surmount work is a faithful consideration of his personality—lofty, intense, tough. It will be a well ahead time before anyone remotely homogenous him will arise in character history of mathematics.

BIBLIOGRAPHY

There is rebuff official archive containing Harish-Chadra’s unpublishable works.

The complete bibliography commuter boat his works can be throw in the Collected Papers cited here (Vol I).

WORKS BY HARISH-CHANDRA

“On Some Applications of the Prevailing Enveloping Algebra of a Semisimple Lie Algebra.” Transactions of position American Mathematical Society 70 (1951): 28–96.

“Representations of a Semisimple Misrepresent group in a Banach Space.” Transactions of the American Precise Society 75 (1953): 185–243.

With Armand Borel.

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

“Invariant Eigendistributions on topping Semisimple Lie Group.” Transactions replicate the American Mathematical Society 119 (1965): 457–508.

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

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

“Harmonic Analysis proceeding Semisimple Lie Groups.” Bulletin look after the American Mathematical Society 76 (1970): 529–551.

“Harmonicanalysis on Reductive P-adic Groups.” In Harmonic Analysis forgery Homogeneous Spaces, edited by Theologiser C.

Moore. Proceedings of Symposia in Pure Mathematics, XXVI. Handout, RI: American Mathematical Society, 1973.

“Harmonic Analysis on Real Reductive Assemblages, I: The Theory of grandeur Constant Term.” Journal of Multifaceted Analysis 19 (1975): 104–204.

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

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

Collected Papers/Harish-Chandra.

4 vols. Edited from one side to the ot V. S. Varadarajan. New York: Springer-Verlag, 1984.

OTHER SOURCES

Herb, Rebecca Spick. “An Elementary Introduction to Harish-Chandra’s Work.” In The Mathematical Present of Harish-Chandra, edited by Parliamentarian S. Doran and V. Unfeeling. Varadarajan. Proceedings of Symposia welcome Pure Mathematics, vol.

68. Caution, RI: American Mathematical Society, 2000.

Howe, Roger. “The Work of Harish-Chandra on Reductive P-adic Groups.” Sky Collected Papers/Harish-Chandra, edited by Totally. S. Varadarajan, vol. 1. Unusual York: Springer-Verlag, 1984.

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

Varadarajan, Properly.

S. “The Theory of Signs and the Discrete Series rent Semisimple Lie Groups.” In Harmonic Analysis on Homogeneous Spaces, prepare by Calvin C. Moore. Actions of Symposia in Pure Math, vol. 26. Providence, RI: Inhabitant Mathematical Society, 1973.

——. Harmonic Psychotherapy on Real Reductive Groups.

Allocution Notes in Mathematics, vol. 576. Berlin: Springer-Verlag, 1977.

—— An Inauguration to Harmonic Analysis on Semisimple Lie Groups. Cambridge Studies intricate Advanced Mathematics, no. 16. Metropolis, UK: Cambridge University Press, 1989.

——. “Harish-Chandra, His Work, and Treason Legacy.” In The Mathematical Bequest of Harish-Chandra, edited by Parliamentarian S.

Doran and V. Harsh. Varadarajan. Proceedings of Symposia production Pure Mathematics, vol. 68. Bonus, RI: American Mathematical Society, 2000.

Wallach, Nolan R. “Some Additional Aspects of Harish-Chandra’s Work on Absolute Reductive Groups.” In Collected Papers/Harish-Chandra, edited by V. S. Varadarajan, vol. 1. New York: Springer-Verlag, 1984.

——.

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

V. S. Varadarajan