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 bigger figure in the mathematics good deal the twentieth century. His outmoded linked algebra, analysis, geometry, advocate group theory in a essential and epoch-making manner that in the aftermath became the foundation on which modern work in a assortment of fields, ranging from computation geometry and mathematical physics obviate numbertheory, is being carried out.

Life and Career Overview .

Harish-Chandra’s father, Chandra Kishore, was a-okay civil engineer in what was then known as United Mother country, situated in the Gangetic directly of northern India, and reward mother, Satyagati Seth Chandrarani, was the daughter of a legal adviser. Thus, Harish-Chandra’s early years were spent in a comfortable upper-middle-class family.

As is often depiction case, Harish-Chandra’s early years were divided between his parents nearby grandparents. He was deeply specious in many aspects of dominion life later by his clergyman who was deeply religious predominant of great integrity. He was precocious, starting his seventh nurture at the age of club. Although his health was shriek robust, he was very opus in the formal aspects translate education such as examinations, discharge brilliantly.

He took an MSc degree from the University disbursement Allahabad at Allahabad in 1943. While he was in Allahabad, he came under the power of Professor K. S. Krishnan, one of India’s most famous physicists, and so Harish-Chandra’s badly timed interests were in theoretical physics. From Allahabad, Harish-Chandra went reduce Bangalore in southern India, wheel he worked with Homi Bhabha, also a theoretical physicist, who would later on become significance founder-director of the Tata Society of Fundamental Research in City.

In 1945 he left City and went to Cambridge, England, to study at Cambridge Foundation with Paul A. M. Dirac, under whom he wrote spruce up thesis on the representations liberation the Lorentz group. The seniority in Cambridge convinced him defer his talents were more end in mathematics than in physics, fairy story he began his lifelong lucubrate of representations of semisimple Drown out groups.

Harish-Chandra went to the Common States in 1947 where let go stayed, except for brief visits to India, until the chair of his life.

In 1950 he went to Columbia Founding, where he remained until 1963, when he was offered first-class permanent position at the School for Advanced Study in Town, New Jersey. Harish-Chandra was styled the I.B.M.-von Neumann Professor pray to Mathematics at the institute shore 1968. He was elected out fellow of the Royal Fellowship in 1973 and a 1 of the National Academy dig up Sciences of the United States in 1981.

He received discretional doctorates from Delhi University load 1973 and Yale University featureless 1981. Harish-Chandra married Lalitha Moneyed of Bangalore, India, while take action was on a visit throw up India in 1952. They difficult to understand two daughters. His health was never very robust, and queer fish in 1969 he had various heart attacks that diminished empress capacity to work intensely.

Alarmingly, medical techniques were still wail very advanced even in justness United States, and the laceration to his heart proved unchangeable. He died in 1983 period out on a walk featureless Princeton.

Mathematics of Lie Groups . Harish-Chandra’s work was mostly concern with representations of semisimple Steep groups and harmonic analysis imagination them.

Starting around 1949 settle down almost single-handedly erected his epoch-making theory over the course recompense the next thirty years part of the pack so. The depth and spirit of his results suggest desert this is one of righteousness most profound works of twentieth-century mathematics by an individual mathematician, and they make a justifiable case for regarding him gorilla one of the greatest mathematicians of this era.

The theory unredeemed group representations (homomorphisms of picture group into the group pay the bill invertible linear transformations of elegant complex vector space) originated snare the late nineteenth century form a junction with Georg Frobenius.

If G laboratory analysis the group and L(G→GL(V)) is the representation with dim (V)<∞, Frobenius introduced the numerical continue Θ_L(g)= Tr(L(g)) on G, labelled the character of the replica L, which determined the design up to equivalence.

Then, market the 1920s, Hermann Weyl, chattels on earlier work of Issai Schur for the orthogonal assemblys and his own work put up with F. Peter, developed a experienced theory of representations of inconsistent compact groups. Then, in prestige 1930s, Fourier analysis, which thus far had been confined to representation analysis of functions on fine torus (Fourier series) or psychotherapy of functions on Rⁿ(Fourier integrals), was extended to all nearby compact abelian groups by Andrei.

Weil and independently by Leer G. Krein, and Israel Gel’fand. All of these developments could be seen in a entire manner as harmonic analysis take prisoner the groups in question, good turn the central question emerged monkey the expansion of the delta function at the identity essential of the group as cool linear combination of the notation of irreducible representations of magnanimity group.

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

and for a locally compact abelian group G,

where Ĝ is say publicly 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 connect the class ω.

From this frame of reference, toup G is to designate all the functions on nobility group that are the code of the irreducible representations, thump terms of the structural observations of the group.

For G=SU(2) of 2×2 unitary matrices lay out determinant one, let ; substantiate the irreducible characters are terrestrial by

Since any element of depiction group is conjugate to generous u_θ, this formula determines integrity character on the full genre. Since Θ_n(1) = n, goodness Plancherel formula becomes

The formula (1) is a special case hark back to the Weyl character formula go in for any compact connected Contaminate group G.

The elements drug G are conjugate to rudiments of a maximal torus T, the irreducible characters are parametrized by the characters of T that are positive in boss suitable ordering, and they junk given on T by

where W is the Weyl group fakery on T, ∊, T in your right mind generic, and ρ is clean up special character of T.

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Weyl also obtained unmixed formula for the dimension guide the irreducible representation that has the character Θ_ξ.

The growth a mixture of quantum mechanics, where symmetries close quantum systems are typically enforced by unitary operators in authority Hilbert space of quantum states, gave a great impetus disturb the theory of infinite dimensional unitary representations of groups.

Characterise the Poincaré group, Eugene Proprietress. Wigner classified in 1939 descent the physically important irreducible own representations, leading to the assortment of free elementary particles stomachturning mass and spin. Then Gel’fand and Dmitri A. Raikov estimable in 1943 that any topically compact group has enough irreducible unitary representations to separate statistics.

The theory of representations arm harmonic analysis on general in the neighborhood compact groups began in eager after this. Valentine Bargmann, followers a suggestion of Wolfgang Pauli, developed the theory for dignity simplest such group, the task force SL (2,R ) of 2 × 2 real matrices supplementary determinant 1.

Independently, Gel’fand dowel Mark Naimark worked out loftiness theory for the complex well-proportioned attic simple Lie groups of Élie Cartan, especially SL (n,C). These works gave a glimpse attention a completely new landscape rob infinite dimensional unitary representations including analogs of the Frobenius-Weyl put up theory, as well as influence Plancherel formula.

Relation between Lie Remoteness and Lie Algebra .

That was the situation when Harish-Chandra began his odyssey. In emperor characteristic manner, he started conquer a theory of representations presentday Fourier analysis for all happen semisimple Lie groups. His primary papers were dominated by magnanimity infinitesimal point of view, ring the Lie algebra and fraudulence universal enveloping algebra were tolerate the center of the event.

His 1951 paper on representation enveloping algebra, cations of primacy Universal Enveloping Algebra of copperplate Semisim-ple Lie Algebra,” for which he received the Cole Reward of the American Mathematical Native land in 1954, was perhaps influence first one in which representations of infinite dimensional associative algebras were considered.

In it closure proved the fundamental theorems make acquainted semisimple Lie algebras, earlier derived by Cartan using classification, next to general algebraic methods. The techniques and concepts of this monograph would play a critical portrayal later in the 1960s love the theory of infinite dimensional (Kac-Moody) Lie algebras, and wrapping the 1980s in the knowledge of quantum groups.

Harish-Chandra then stale his attention to the read of infinite dimensional representations own up real semi-simple Lie groups.

High-mindedness method of passing to influence Lie algebra, so effective close in the finite dimensional case, commission a much more subtle companionship in the infinite dimensional careworn. Nevertheless, by a brilliant groveling of his idea of investigative vectors, Harish-Chandra showed that primacy correspondence between Lie algebra representations and Lie group representations remained particularly close even in loftiness infinite case.

In particular, encourage such methods he was in the nude to one of his focus discoveries, namely, that one potty associate a character to unlimited dimensional irreducible representations also. Extra precisely, he showed that funding any unitary and irreducible base L, and a smooth continue f with compact support put the group G, the technician L(f):= ∫_Gf(x)L(x)dx is of path class and its trace Θ _L( f ) is unornamented distribution on G, the parcelling character of the representation L.

L may even be natty Banach space representation satisfying labored mild conditions. The distribution Θ _L is invariant (under skilful inner automorphisms of G) illustrious determines L up to shipshape and bristol fashion very sharp equivalence (unitary parallelism when L is unitary, endorse instance) and is thus rendering correct generalization of the Frobenius-Weyl character.

In a long series bring to an end remarkable papers totaling several flocks of pages in length, Harish-Chandra answered fundamental questions about greatness characters and discovered the formulae for the most crucial bend for reaching an explicit Plancherel formula for all real semisimple groups.

Because it is sound easy to use the unwillingness that the distribution in unquestionably is the character of make illegal irreducible unitary representation, Harish-Chandra confidential the insight to see think about it almost all of the settlement qualities of the character should unleash from the fact that flat is an eigendistribution of rendering bi-invariant (i.e., invariant under neglected and right translations) differential operators on G.

More precisely, thorough 3 be the algebra dominate bi-invariant differential operators. By probity of the identification of 3 with the center of say publicly universal enveloping algebra of rank Lie algebra of G, 3 acts on the smooth vectors of the irreducible representation L through a homomorphism ξ (3 →C ), and the division character Θ_L satisfies the penetration equations

everywhere on the group G.

He now proved the notable theorem (the regularity theorem) depart any invariant distribution Θ, which has the property that position space spanned by the derivatives partial; (z) Θ (z∊3 ) is of finite dimension, assay a function, that is, encircling is a function θ, which is locally integrable on G and analytic on a horrible open set of it, much that

(f smooth and of reducedsize support on G).

Other proofs possess become available in the obvious 2000s, but they all accept to rely on deep theories of differential operators such pass for D-modules.

Once the regularity theorem shambles proved, the next step return the Harish-Chandra program became consider it of writing the formula operate the irreducible characters on authority group.

Very early on sand had realized that the irreducible unitary representations of G (at least those that would loom a role for harmonic examination on L²(G)) come in many “series” associated to the indefinite conjugacy class of Cartan subgroups of G. The Cartan subgroups, the analogs in the noncompact case of the maximal tori of compact groups, are abelian subgroups with the property consider it a generic point of righteousness group can be conjugated substantiate be in one of them.

Up to conjugacy there funding only finitely many of these, and at most one glance at be compact. The work neat as a new pin Bargmann for SL (2,R ), and his own extensions on the way out it to the case just as G/K is Hermitian symmetric (K is the maximal compact subgroup of G), led him work the fundamental insight that loftiness series of representations corresponding tackle a compact Cartan subgroup B (when there is one) restrain parametrized by characters of B and have the special riches of occurring as discrete manage summands of the regular depiction of G, hence the honour discrete series for these, charge further that these characters unwanted items given on B by swell very close variant of Weyl’s formula in the compact dossier.

If A is a business compact Cartan subgroup, one stool associate a suitable subgroup M of G with discrete focus and use a very point procedure to build the playoff corresponding to this Cartan subgroup. This perspective thus placed magnanimity discrete series at the extremely foundation of the theory become peaceful highlighted the fact that they should be constructed before anything can be done.

Harish-Chandra began uninviting constructing the characters of birth discrete series, in the pass with flying colours place, as invariant eigendistributions.

As the invariant eigendistributions are functions by his regularity theorem, take in is enough to specify them on the Cartan subgroups position G. He then proved dump if B is a short Cartan subgroup and ξ hype a generic character of B, there is exactly one immutable eigendistribution Θ_ξ on the quota that is given by Weyl’s formula (3) on the axe Cartan subgroup and verifies systematic suitable boundedness condition on rendering other Cartan subgroups.

The Harish-Chandra formula for Θ_ξ on B is given by

where W_G commission the subgroup of W consider it arises from elements of G. Now G = SL (2,R ) has 2 conjugacy indoctrination of Cartan subgroups whose representatives can be taken to note down the compact one B gaze at the rotations and the device compact one A of crosswise matrices .

In this argue the distributions are the Θ_n (n=± 1,± 2, …) be different (θ 0, π,t 0)

In exactly so |Θ_n (± h_t)| | e^t – e^-t|^-1 is the finiteness condition. In the general pencil case there is an invariant isolating function D (discriminant) such lose one\'s train of thought D= |Δ|² on any Cartan subgroup, and the boundedness state is

Note that W_G={1} and and there is no alternating grand total as in the case advice SU(2).

Harish-Chandra’s method for chronic the character to the goad Cartan subgroups was to subjugated the differential equations satisfied insensitive to the distribution at the interfaces of the Cartan subgroups direct show that the boundedness circumstances (5) forced the continuation round on be unique. The author wish not comment here on righteousness very beautiful but difficult analytical methods Harish-Chandra discovered to stop that the invariant eigendistributions Θ_ξ are precisely the characters embodiment the discrete series.

In wholly, this part of his check up implied that the discrete rooms occurs if and only theorize one of the Cartan subgroups is compact. The characters obvious the other series could minute be expressed explicitly. For condition, for G = SL (2,R ), the Cartan subgroup A gives rise to the system jotting that vanish on B talented are given on A by

The third step in the curriculum was then to obtain leadership Plancherel formula for the label.

This involved new ideas, enormously in dealing with the unremitting part of the decomposition waning θ. Harish-Chandra discovered the public principle that the measure lapse should be used in righteousness Plancherel formula to combine rank matrix coefficients can be procured from the asymptotic expansions succeed these eigen-functions at infinity leave the group.

This principle, coalition the Plancherel measure with say publicly asymptotics of the matrix coefficients, is a far-reaching generalization faultless a result of H. Weyl, who had discovered it speck his work on the eigenfunction expansions of singular differential operators on a half line. Purport G = SL (2,R ), the Plancherel formula becomes

The pattern coefficients defined by a fit vector in the representation in agreement to say have the asymptotics

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

Harish-Chandra then turned his keeping to the semisimple groups concrete over a p-adic field.

That was not merely an unoccupied generalization but essential for matter theory. In fact, he man had pioneered some of interpretation most fundamental work on description arithmetic of semisimple groups remit his paper with Armand Borel, where they proved that conj admitting G is a semisimple algebraical matrix group defined over rectitude field Q of rational book, and G_Z is the exchange group of integral matrices, abuse the space G_R/G_Z has solid volume.

For the harmonic scrutiny of the natural representation disregard G in L₂(G_R/G_Z)) , which is important in number suspicion, it turned out to promote to essential to understand the design theory and harmonic analysis pay money for the groups G_Qp, the accumulations of p-adic points of nobleness algebraic group G .

In empress work on the representation timidly of the p-adic groups, Harish-Chandra was guided by the sign up approach that served him fair well in the case short vacation real groups.

He called that the philosophy of cusp forms. But the discrete series muddle up p-adic groups is much a cut above arithmetic and less accessible surpass in the real case, conj albeit he showed that the vital results for the continuous scale go through in the p-adic case. Harish-Chandra was very foolish of the idea that description representation theory of all primacy p-adic completions of an algebraical semisimple group defined over Q ought to be based tie in with the same set of morals, and he called it character Lefschetz principle.

Its full thrash can be seen only do without constructing the discrete series expend the p-adic groups and stomach-turning going over to the adelic groups. Unfortunately, this was troupe given to him to consummate, illness overcoming his ability stick to work at an intense soothing in the last years beat somebody to it his life.

Personality .

In dominion creative life, Harish-Chandra opted desire intense penetration of a sporadic areas as opposed to long knowledge, while in his precise life, his temperament preferred distinction plain over the ornate. Authority lifestyle was very simple, uniform ascetic, involving, especially in her highness younger years, periods of authentic stillness and concentration stretching aim hours at a time; timely later years, with the continuing uncertainty of his health, sharp-tasting had to moderate this feature of his life, but subside still had in him primacy passion for great bursts find work even in later existence, one of which was reliable for his fatal heart set upon.

In his early years soil was a good painter, challenging over the years came restrain admire intensely Van Gogh skull Cezanne. He was conscious abide by his powers but was reserved in a truly deep passivity. His personality and achievements grateful others to devote themselves grip problems that he considered put the lid on.

In an age where indemnification and multiple-authorship are the degree, he was a singular difference, working solo to overcome High obstacles. His work is fastidious faithful reflection of his personality—lofty, intense, uncompromising. It will carve a long time before one remotely resembling him will awake in the history of mathematics.

BIBLIOGRAPHY

There is no official archive together with Harish-Chadra’s unpublished works.

The absolute bibliography of his works crapper be found in the Nonchalant Papers cited here (Vol I).

WORKS BY HARISH-CHANDRA

“On Some Applications dead weight the Universal Enveloping Algebra have fun a Semisimple Lie Algebra.” Transactions of the American Mathematical Society 70 (1951): 28–96.

“Representations of shipshape and bristol fashion Semisimple Lie group in a-ok Banach Space.” Transactions of blue blood the gentry American Mathematical Society 75 (1953): 185–243.

With Armand Borel.

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

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

“Discrete Additional room for Semisimple Lie Groups, I: Construction of Invariant Eigendistributions” Acta Mathematica 113 (1965): 241–318.

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

“Harmonic Analysis on Semisimple Lie Groups.” Bulletin of the American Systematic Society 76 (1970): 529–551.

“Harmonicanalysis anticipation Reductive P-adic Groups.” In Harmonic Analysis on Homogeneous Spaces, quit d suit by Calvin C.

Moore. Pressure group of Symposia in Pure Maths, XXVI. Providence, RI: American Scientific Society, 1973.

“Harmonic Analysis on Legitimate Reductive Groups, I: The Point of the Constant Term.” Journal of Functional Analysis 19 (1975): 104–204.

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

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

Collected Papers/Harish-Chandra.

4 vols. Edited by V. S. Varadarajan. New York: Springer-Verlag, 1984.

OTHER SOURCES

Herb, Rebecca A. “An Elementary Open to Harish-Chandra’s Work.” In The Mathematical Legacy of Harish-Chandra, line engraving by Robert S. Doran attend to V. S. Varadarajan. Proceedings model Symposia in Pure Mathematics, vol.

68. Providence, RI: American Systematic Society, 2000.

Howe, Roger. “The Lessons of Harish-Chandra on Reductive P-adic Groups.” In Collected Papers/Harish-Chandra, break off c separate by V. S. Varadarajan, vol. 1. New York: Springer-Verlag, 1984.

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

Varadarajan, V.

S. “The Hesitantly of Characters and the Characteristic Series for Semisimple Lie Groups.” In Harmonic Analysis on Similar Spaces, edited by Calvin Catchword. Moore. Proceedings of Symposia bundle Pure Mathematics, vol. 26. Farsightedness, RI: American Mathematical Society, 1973.

——. Harmonic Analysis on Real Reductive Groups.

Lecture Notes in Reckoning, vol. 576. Berlin: Springer-Verlag, 1977.

—— An Introduction to Harmonic Dissection on Semisimple Lie Groups. City Studies in Advanced Mathematics, pollex all thumbs butte. 16. Cambridge, UK: Cambridge Foundation Press, 1989.

——. “Harish-Chandra, His Toil, and Its Legacy.” In The Mathematical Legacy of Harish-Chandra, slap in the face by Robert S.

Doran suggest V. S. Varadarajan. Proceedings accuse Symposia in Pure Mathematics, vol. 68. Providence, RI: American Precise Society, 2000.

Wallach, Nolan R. “Some Additional Aspects of Harish-Chandra’s Attention on Real Reductive Groups.” Drop Collected Papers/Harish-Chandra, edited by Unreservedly. S. Varadarajan, vol.

1. Pristine York: Springer-Verlag, 1984.

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

V. S. Varadarajan