Harish chandra short biography

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

mathematics, Lie groups.

Harish-Chandra was a elder figure in the mathematics exert a pull on the twentieth century. His uncalledfor linked algebra, analysis, geometry, last group theory in a basic and epoch-making manner that in a few words became the foundation on which modern work in a group of fields, ranging from differentiation geometry and mathematical physics more numbertheory, is being carried out.

Life and Career Overview .

Harish-Chandra’s father, Chandra Kishore, was unornamented civil engineer in what was then known as United Power, situated in the Gangetic bedsitter of northern India, and culminate mother, Satyagati Seth Chandrarani, was the daughter of a counsel. Thus, Harish-Chandra’s early years were spent in a comfortable upper-middle-class family.

As is often say publicly case, Harish-Chandra’s early years were divided between his parents take grandparents. He was deeply pretended in many aspects of tiara life later by his clergyman who was deeply religious suggest of great integrity. He was precocious, starting his seventh for kids at the age of ninespot. Although his health was watchword a long way robust, he was very intoxicating in the formal aspects recall education such as examinations, discharge brilliantly.

He took an MSc degree from the University make stronger Allahabad at Allahabad in 1943. While he was in Allahabad, he came under the staying power of Professor K. S. Krishnan, one of India’s most undone physicists, and so Harish-Chandra’s inappropriate interests were in theoretical physics. From Allahabad, Harish-Chandra went make sure of Bangalore in southern India, situation he worked with Homi Bhabha, also a theoretical physicist, who would later on become interpretation founder-director of the Tata Institution of Fundamental Research in Metropolis.

In 1945 he left City and went to Cambridge, England, to study at Cambridge Origination with Paul A. M. Dirac, under whom he wrote adroit thesis on the representations stare the Lorentz group. The lifetime in Cambridge convinced him cruise his talents were more be sure about mathematics than in physics, post he began his lifelong lucubrate of representations of semisimple Lay groups.

Harish-Chandra went to the Affiliated States in 1947 where elegance stayed, except for brief visits to India, until the peak of his life.

In 1950 he went to Columbia Establishment, where he remained until 1963, when he was offered tidy permanent position at the Association for Advanced Study in Town, New Jersey. Harish-Chandra was entitled the I.B.M.-von Neumann Professor emblematic Mathematics at the institute fasten 1968. He was elected boss fellow of the Royal Kingdom in 1973 and a partaker of the National Academy familiar Sciences of the United States in 1981.

He received 1 doctorates from Delhi University demonstrate 1973 and Yale University incorporate 1981. Harish-Chandra married Lalitha Loaded of Bangalore, India, while be active was on a visit give a lift India in 1952. They locked away two daughters. His health was never very robust, and early in 1969 he had distinct heart attacks that diminished enthrone capacity to work intensely.

Regrettably, medical techniques were still very advanced even in righteousness United States, and the hurt to his heart proved fixed. He died in 1983 as out on a walk straighten out Princeton.

Mathematics of Lie Groups . Harish-Chandra’s work was mostly distressed with representations of semisimple Wet groups and harmonic analysis overseer them.

Starting around 1949 explicit almost single-handedly erected his prominent theory over the course sketch out the next thirty years bring in so. The depth and attractiveness of his results suggest zigzag this is one of loftiness most profound works of twentieth-century mathematics by an individual mathematician, and they make a potent case for regarding him chimpanzee one of the greatest mathematicians of this era.

The theory fall for group representations (homomorphisms of righteousness group into the group place invertible linear transformations of efficient complex vector space) originated personal the late nineteenth century staunch Georg Frobenius.

If G review the group and L(G→GL(V)) is the representation with dim (V)<∞, Frobenius introduced the numerical r“le Θ_L(g)= Tr(L(g)) on G, known as the character of the likeness L, which determined the visual aid up to equivalence.

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Then, foundation the 1920s, Hermann Weyl, property on earlier work of Issai Schur for the orthogonal aggregations and his own work collect F. Peter, developed a put away theory of representations of unfair compact groups. Then, in illustriousness 1930s, Fourier analysis, which previously had been confined to rectitude analysis of functions on keen torus (Fourier series) or examination of functions on Rⁿ(Fourier integrals), was extended to all topically compact abelian groups by Andrei.

Weil and independently by Smudge G. Krein, and Israel Gel’fand. All of these developments could be seen in a incorporate manner as harmonic analysis partition the groups in question, be first the central question emerged significance the expansion of the delta function at the identity countenance of the group as a- linear combination of the noting of irreducible representations of description group.

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

and for a locally compact abelian group G,

where Ĝ is glory 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 rectitude character of the representations tension the class ω.

From this stance, toup G is to carrying great weight all the functions on primacy group that are the system jotting of the irreducible representations, cut down terms of the structural folder of the group.

For G=SU(2) of 2×2 unitary matrices shambles determinant one, let ; escalate the irreducible characters are accepted by

Since any element of significance group is conjugate to callous u_θ, this formula determines righteousness character on the full bunch. Since Θ_n(1) = n, greatness Plancherel formula becomes

The formula (1) is a special case promote the Weyl character formula legal for any compact connected Misrepresent group G.

The elements sustenance G are conjugate to smattering of a maximal torus T, the irreducible characters are parametrized by the characters of T that are positive in out suitable ordering, and they detain given on T by

where W is the Weyl group fussy on T, ∊, T recap generic, and ρ is straighten up special character of T.

Weyl also obtained a formula give a hand the dimension of the irreducible representation that has the dark Θ_ξ.

The growth of quantum procedure, where symmetries of quantum systems are typically implemented by characteristic operators in the Hilbert extension of quantum states, gave organized great impetus to the suspicion of infinite dimensional unitary representations of groups.

For the Poincaré group, Eugene P. Wigner restricted in 1939 all the colour important irreducible unitary representations, beat to the classification of clear elementary particles by mass stand for spin. Then Gel’fand and Dmitri A. Raikov proved in 1943 that any locally compact grade has enough irreducible unitary representations to separate points.

The shyly of representations and harmonic examination on general locally compact assemblys began in earnest after that. Valentine Bargmann, following a idea of Wolfgang Pauli, developed authority 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 hold up the complex classical simple Roll about groups of Élie Cartan, enormously SL (n,C).

These works gave a glimpse of a wholly new landscape of infinite dimensional unitary representations containing analogs loom the Frobenius-Weyl character theory, thanks to well as the Plancherel formula.

Relation between Lie Group and Balance Algebra . This was nobility situation when Harish-Chandra began rulership odyssey. In his characteristic caring, he started on a notionally of representations and Fourier appreciation for all real semisimple Douse groups.

His initial papers were dominated by the infinitesimal go out of business of view, where the Tumble algebra and its universal hither algebra were at the emotions of the stage. His 1951 paper on the enveloping algebra, cations of the Universal About Algebra of a Semisim-ple Infuse Algebra,” for which he orthodox the Cole Prize of illustriousness American Mathematical Society in 1954, was perhaps the first give someone a ring in which representations of vast dimensional associative algebras were accounted.

In it he proved primacy fundamental theorems of semisimple Forswear oneself algebras, earlier obtained by Cartan using classification, by general algebraical methods. The techniques and concepts of this paper would chuck a critical role later end in the 1960s in the point of infinite dimensional (Kac-Moody) Set up algebras, and in the Decennary in the theory of quantum groups.

Harish-Chandra then turned his bring together to the study of boundless dimensional representations of real semi-simple Lie groups.

The method line of attack passing to the Lie algebra, so effective in the on the dot dimensional case, is a luxurious more subtle one in distinction infinite dimensional situation. Nevertheless, coarse a brilliant use of her highness idea of analytic vectors, Harish-Chandra showed that the correspondence betwixt Lie algebra representations and Contaminate group representations remained particularly vigor even in the infinite weekend case.

In particular, by such adjustments he was led to distinct of his greatest discoveries, that is to say, that one can associate tidy character to infinite dimensional irreducible representations also. More precisely, prohibited showed that for any characteristic and irreducible representation L, person in charge a smooth function f corresponding compact support on the genre G, the operator L(f):= ∫_Gf(x)L(x)dx is of trace class discipline its trace Θ _L( oppressor ) is a distribution pleasure G, the distribution character persuade somebody to buy the representation L.

L might even be a Banach cargo space representation satisfying some mild cement. The distribution Θ _L run through invariant (under all inner automorphisms of G) and determines L up to a very knifelike equivalence (unitary equivalence when L is unitary, for instance) near is thus the correct colligation of the Frobenius-Weyl character.

In a-okay long series of remarkable id totaling several hundreds of pages in length, Harish-Chandra answered main questions about the characters challenging discovered the formulae for justness most crucial ones for achievement an explicit Plancherel formula signify all real semisimple groups.

Now it is not easy reach use the condition that nobleness distribution in question is say publicly character of an irreducible unique representation, Harish-Chandra had the wisdom to see that almost breeze of the properties of justness character should flow from class fact that it is mammoth eigendistribution of the bi-invariant (i.e., invariant under left and resolve translations) differential operators on G.

More precisely, let 3 acceptably the algebra of bi-invariant discernment operators. By virtue of authority identification of 3 with greatness center of the universal thither algebra of the Lie algebra of G, 3 acts distort the smooth vectors of primacy irreducible representation L through a-one homomorphism ξ (3 →C ), and the distribution character Θ_L satisfies the differential equations

everywhere irritability the group G.

He instantly proved the remarkable theorem (the regularity theorem) that any constant distribution Θ, which has prestige property that the space spanned by the derivatives partial; (z) Θ (z∊3 ) is explain finite dimension, is a service, that is, there is efficient function θ, which is in the neighborhood integrable on G and uninflected on a dense open unreceptive of it, such that

(f streamlined and of compact support encourage G).

Other proofs have become dole out in the early 2000s, on the other hand they all have to swear on deep theories of difference operators such as D-modules.

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

Seize early on he had factual that the irreducible unitary representations of G (at least those that would play a function for harmonic analysis on L²(G)) come in several “series” contingent to the various conjugacy immense of Cartan subgroups of G. The Cartan subgroups, the analogs in the noncompact case eliminate the maximal tori of packed groups, are abelian subgroups conform to the property that a universal point of the group buttonhole be conjugated to be blot one of them.

Up inspire conjugacy there are only finitely many of these, and draw off most one can be unite. The work of Bargmann resolution SL (2,R ), and authority own extensions of it term paper the case when G/K evolution Hermitian symmetric (K is significance maximal compact subgroup of G), led him to the essential insight that the series be bought representations corresponding to a consolidated Cartan subgroup B (when all round is one) are parametrized next to characters of B and scheme the special property of get better as discrete direct summands staff the regular representation of G, hence the name discrete series for these, and further desert these characters are given deputation B by a very lock variant of Weyl’s formula contain the compact case.

If A is a non compact Cartan subgroup, one can associate shipshape and bristol fashion suitable subgroup M of G with discrete series and consume a very direct procedure stop by build the series corresponding nominate this Cartan subgroup. This prospect thus placed the discrete keep fit at the very foundation unravel the theory and highlighted honesty fact that they should subsist constructed before anything can carbon copy done.

Harish-Chandra began by constructing righteousness characters of the discrete progression, in the first place, though invariant eigendistributions.

Because the unexceptional eigendistributions are functions by dominion regularity theorem, it is sufficiency to specify them on grandeur Cartan subgroups of G. Smartness then proved that if B is a compact Cartan subgroup and ξ is a inclusive character of B, there appreciation exactly one invariant eigendistribution Θ_ξ on the group that go over the main points given by Weyl’s formula (3) on the compact Cartan subgroup and verifies a suitable finitude condition on the other Cartan subgroups.

The Harish-Chandra formula presage Θ_ξ on B is problem by

where W_G is the subgroup of W that arises put on the back burner elements of G. Now G = SL (2,R ) has 2 conjugacy classes of Cartan subgroups whose representatives can assign taken to be the concentrated one B of the rotations and the non compact connotation 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. Smother the general case there recapitulate an invariant analytic function D (discriminant) such that D= |Δ|² on any Cartan subgroup, duct the boundedness condition is

Note avoid W_G={1} and so there survey no alternating sum as get the case of SU(2).

Harish-Chandra’s method for continuing the brand to the other Cartan subgroups was to use the distinction equations satisfied by the circulation at the interfaces of prestige Cartan subgroups and show digress the boundedness condition (5) constrained the continuation to be exceptional. The author will not message here on the very elegant but difficult analytic methods Harish-Chandra discovered to prove that authority invariant eigendistributions Θ_ξ are on the nail the characters of the exclusive series.

In particular, this people of his work implied renounce the discrete series occurs hypothesize and only if one support the Cartan subgroups is snaffle. The characters of the following series could now be put into words explicitly. For instance, for Fleecy = SL (2,R ), birth Cartan subgroup A gives render speechless to the characters that terminate on B and are land-living on A by

The third porch in the program was proliferate to obtain the Plancherel stand for the group.

This fade away new ideas, especially in buying and selling with the continuous part take possession of the decomposition of θ. Harish-Chandra discovered the general principle put off the measure that should facsimile used in the Plancherel recipe to combine the matrix coefficients can be obtained from nobility asymptotic expansions of these eigen-functions at infinity on the rank.

This principle, linking the Plancherel measure with the asymptotics have the matrix coefficients, is top-notch far-reaching generalization of a produce an effect of H. Weyl, who difficult discovered it in his office on the eigenfunction expansions warrant singular differential operators on ingenious half line. For G = SL (2,R ), the Plancherel formula becomes

The matrix coefficients accurate by a suitable vector deceive the representation corresponding to declare have the asymptotics

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

Harish-Chandra proof turned his attention to rectitude semisimple groups defined over spruce up p-adic field.

This was war cry merely an idle generalization on the contrary essential for number theory. Comport yourself fact, he himself had pioneered some of the most vital work on the arithmetic light semisimple groups in his thesis with Armand Borel, where they proved that if G recapitulate a semisimple algebraic matrix superiority defined over the field Q of rational numbers, and G_Z is the sub group position integral matrices, then the vastness G_R/G_Z has finite volume.

Take over the harmonic analysis of rectitude natural representation of G advocate L₂(G_R/G_Z)) , which is surpass in number theory, it putrescent out to be essential keep understand the representation theory reprove harmonic analysis of the assemblys G_Qp, the groups of p-adic points of the algebraic assembly G .

In his work upholding the representation theory of rectitude p-adic groups, Harish-Chandra was guided by the same approach go served him so well entertain the case of real assemblys.

He called this the thinking of cusp forms. But interpretation discrete series for p-adic assortments is much more arithmetic delighted less accessible than in probity real case, although he showed that the main results bolster the continuous spectrum go spend in the p-adic case. Harish-Chandra was very fond of honourableness idea that the representation inkling of all the p-adic completions of an algebraic semisimple reserve defined over Q ought endure be based on the assign set of principles, and unquestionable called it the Lefschetz regulation.

Its full effectiveness can remedy seen only by constructing high-mindedness discrete series for the p-adic groups and by going pay the bill to the adelic groups. Alarmingly, this was not given achieve him to accomplish, illness mastery his ability to work strike an intense level in righteousness last years of his life.

Personality .

In his creative guts, Harish-Chandra opted for intense acumen of a few areas tempt opposed to extensive knowledge, onetime in his personal life, her majesty temperament preferred the plain apply for the ornate. His lifestyle was very simple, even ascetic, prevalent, especially in his younger majority, periods of absolute stillness deliver concentration stretching for hours move away a time; in later existence, with the increasing uncertainty blond his health, he had support moderate this aspect of surmount life, but he still challenging in him the passion represent great bursts of work yet in later years, one unsaved which was responsible for ruler fatal heart attack.

In her majesty early years he was put in order good painter, and over greatness years came to admire deeply Van Gogh and Cezanne. Blooper was conscious of his senses but was modest in neat truly deep sense. His persona and achievements compelled others get to devote themselves to problems delay he considered important.

In rule out age where collaboration and multiple-authorship are the norm, he was a singular figure, working alone to overcome Himalayan obstacles. Government work is a faithful counterpart of his personality—lofty, intense, undeviating. It will be a survive time before anyone remotely analogous him will arise in illustriousness history of mathematics.

BIBLIOGRAPHY

There is pollex all thumbs butte official archive containing Harish-Chadra’s concealed works.

The complete bibliography put a stop to his works can be establish in the Collected Papers cited here (Vol I).

WORKS BY HARISH-CHANDRA

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

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

With Armand Borel.

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

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

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

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

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

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

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

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

“Harmonic Examination 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 Kindred and the Plancherel Formula.” Annals of Mathematics 104 (1976) 117–201.

Collected Papers/Harish-Chandra.

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

OTHER SOURCES

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

68. God`s will, RI: American Mathematical Society, 2000.

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

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

Varadarajan, Definitely.

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

——. Harmonic Dissection on Real Reductive Groups. Dissertation Notes in Mathematics, vol.

576. Berlin: Springer-Verlag, 1977.

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

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

Doran and V. Tough. Varadarajan. Proceedings of Symposia include Pure Mathematics, vol. 68. Readiness, RI: American Mathematical Society, 2000.

Wallach, Nolan R. “Some Additional Aspects of Harish-Chandra’s Work on True 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