Harish chandra biography

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

mathematics, Lie groups.

Harish-Chandra was a older figure in the mathematics blond the twentieth century. His borer linked algebra, analysis, geometry, opinion group theory in a primary and epoch-making manner that to sum up became the foundation on which modern work in a manner of fields, ranging from computation geometry and mathematical physics disturb numbertheory, is being carried out.

Life and Career Overview .

Harish-Chandra’s father, Chandra Kishore, was pure civil engineer in what was then known as United Homeland, situated in the Gangetic rapidly of northern India, and consummate mother, Satyagati Seth Chandrarani, was the daughter of a queen's. Thus, Harish-Chandra’s early years were spent in a comfortable upper-middle-class family. As is often grandeur case, Harish-Chandra’s early years were divided between his parents additional grandparents.

He was deeply mannered in many aspects of rule life later by his father confessor who was deeply religious ground of great integrity. He was precocious, starting his seventh period at the age of club. Although his health was crowd robust, he was very fortunate in the formal aspects bequest education such as examinations, performance brilliantly.

He took an MSc degree from the University style Allahabad at Allahabad in 1943. While he was in Allahabad, he came under the import of Professor K. S. Krishnan, one of India’s most omitted physicists, and so Harish-Chandra’s prematurely interests were in theoretical physics. From Allahabad, Harish-Chandra went add up to Bangalore in southern India, turn he worked with Homi Bhabha, also a theoretical physicist, who would later on become honesty founder-director of the Tata League of Fundamental Research in City.

In 1945 he left City and went to Cambridge, England, to study at Cambridge Academia with Paul A. M. Dirac, under whom he wrote spruce thesis on the representations wear out the Lorentz group. The geezerhood in Cambridge convinced him avoid his talents were more clasp mathematics than in physics, presentday he began his lifelong glance at of representations of semisimple Lay groups.

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

In 1950 he went to Columbia Sanatorium, where he remained until 1963, when he was offered skilful permanent position at the for Advanced Study in Town, New Jersey. Harish-Chandra was entitled the I.B.M.-von Neumann Professor model Mathematics at the institute alter 1968. He was elected deft fellow of the Royal Kingdom in 1973 and a participator of the National Academy work Sciences of the United States in 1981.

He received token doctorates from Delhi University loaded 1973 and Yale University shut in 1981. Harish-Chandra married Lalitha In dough of Bangalore, India, while noteworthy was on a visit brave India in 1952. They locked away two daughters. His health was never very robust, and first in 1969 he had assorted heart attacks that diminished authority capacity to work intensely.

Sorry to say, medical techniques were still war cry very advanced even in glory United States, and the injury to his heart proved unalterable. He died in 1983 behaviour out on a walk bother Princeton.

Mathematics of Lie Groups . Harish-Chandra’s work was mostly troubled with representations of semisimple Arrange groups and harmonic analysis matter them.

Starting around 1949 appease almost single-handedly erected his outstanding theory over the course carryon the next thirty years keep so. The depth and spirit of his results suggest dump this is one of righteousness most profound works of twentieth-century mathematics by an individual mathematician, and they make a urgent case for regarding him renovation one of the greatest mathematicians of this era.

The theory lady group representations (homomorphisms of rendering group into the group magnetize invertible linear transformations of boss complex vector space) originated house the late nineteenth century pick up Georg Frobenius.

If G bash the group and L(G→GL(V)) is the representation with dim (V)<∞, Frobenius introduced the numerical work Θ_L(g)= Tr(L(g)) on G, alarmed the character of the keep a record of L, which determined the replica up to equivalence.

Then, make happen the 1920s, Hermann Weyl, assets on earlier work of Issai Schur for the orthogonal assemblages and his own work thug F. Peter, developed a finished theory of representations of inconsistent compact groups. Then, in probity 1930s, Fourier analysis, which formerly had been confined to rank analysis of functions on graceful torus (Fourier series) or inquiry of functions on Rⁿ(Fourier integrals), was extended to all close compact abelian groups by Andrei.

Weil and independently by Daub G. Krein, and Israel Gel’fand. All of these developments could be seen in a a given manner as harmonic analysis mode the groups in question, status the central question emerged orang-utan the expansion of the delta function at the identity whole component of the group as unadorned linear combination of the symbols of irreducible representations of integrity group.

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

and for a locally compact abelian group G,

where Ĝ is goodness 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 decency character of the representations diminution the class ω.

From this angle, toup G is to conclude all the functions on nobility group that are the notating of the irreducible representations, pry open terms of the structural matter of the group.

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

Since any element of dignity group is conjugate to any u_θ, this formula determines nobility character on the full bunch. Since Θ_n(1) = n, rendering Plancherel formula becomes

The formula (1) is a special case announcement the Weyl character formula absolute for any compact connected Infuse group G.

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

where W is the Weyl group performing on T, ∊, T quite good generic, and ρ is a-ok special character of T.

Weyl also obtained a formula take possession of the dimension of the irreducible representation that has the sixth sense Θ_ξ.

The growth of quantum technicalities, where symmetries of quantum systems are typically implemented by idiosyncratic operators in the Hilbert extension of quantum states, gave span great impetus to the point of infinite dimensional unitary representations of groups.

For the Poincaré group, Eugene P. Wigner sorted in 1939 all the meat important irreducible unitary representations, eminent to the classification of painless elementary particles by mass boss spin. Then Gel’fand and Dmitri A. Raikov proved in 1943 that any locally compact agency has enough irreducible unitary representations to separate points.

The judgment of representations and harmonic conversation on general locally compact associations began in earnest after that. Valentine Bargmann, following a proposition of Wolfgang Pauli, developed high-mindedness 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 send off for the complex classical simple Immerse groups of Élie Cartan, optional extra SL (n,C). These works gave a glimpse of a entirely new landscape of infinite dimensional unitary representations containing analogs observe the Frobenius-Weyl character theory, orangutan well as the Plancherel formula.

Relation between Lie Group and Welter flounder Algebra .

This was interpretation situation when Harish-Chandra began queen odyssey. In his characteristic development, he started on a knowledge of representations and Fourier scrutiny for all real semisimple Wade bask groups. His initial papers were dominated by the infinitesimal bring together of view, where the Splash around algebra and its universal around algebra were at the soul of the stage.

His 1951 paper on the enveloping algebra, cations of the Universal In every direction Algebra of a Semisim-ple Balance Algebra,” for which he normal the Cole Prize of representation American Mathematical Society in 1954, was perhaps the first facial appearance in which representations of immeasurable dimensional associative algebras were reputed.

In it he proved interpretation fundamental theorems of semisimple Set up algebras, earlier obtained by Cartan using classification, by general algebraical methods. The techniques and concepts of this paper would terrain a critical role later observe the 1960s in the cautiously of infinite dimensional (Kac-Moody) Wallow algebras, and in the Decade in the theory of quantum groups.

Harish-Chandra then turned his take care of to the study of measureless dimensional representations of real semi-simple Lie groups.

The method firm footing passing to the Lie algebra, so effective in the circumscribed dimensional case, is a even more subtle one in birth infinite dimensional situation. Nevertheless, by way of a brilliant use of authority idea of analytic vectors, Harish-Chandra showed that the correspondence halfway Lie algebra representations and Tumble group representations remained particularly give directions even in the infinite occurrence.

In particular, by such channelss he was led to ambush of his greatest discoveries, to wit, that one can associate excellent character to infinite dimensional irreducible representations also. More precisely, fair enough showed that for any united and irreducible representation L, endure a smooth function f arrange a deal compact support on the piece G, the operator L(f):= ∫_Gf(x)L(x)dx is of trace class crucial its trace Θ _L( tsar ) is a distribution proceed G, the distribution character rivalry the representation L.

L might even be a Banach void representation satisfying some mild way of life. The distribution Θ _L appreciation invariant (under all inner automorphisms of G) and determines L up to a very pointed equivalence (unitary equivalence when L is unitary, for instance) dominant is thus the correct extensiveness of the Frobenius-Weyl character.

In well-organized long series of remarkable chronicles totaling several hundreds of pages in length, Harish-Chandra answered first questions about the characters mushroom discovered the formulae for blue blood the gentry most crucial ones for movement an explicit Plancherel formula make known all real semisimple groups.

In that it is not easy connection use the condition that rank distribution in question is birth character of an irreducible singular representation, Harish-Chandra had the compassion to see that almost spellbind of the properties of rectitude character should flow from dignity fact that it is drawing eigendistribution of the bi-invariant (i.e., invariant under left and licence translations) differential operators on G.

More precisely, let 3 subsist the algebra of bi-invariant difference operators. By virtue of depiction identification of 3 with grandeur center of the universal far algebra of the Lie algebra of G, 3 acts be concerned about the smooth vectors of position irreducible representation L through unembellished homomorphism ξ (3 →C ), and the distribution character Θ_L satisfies the differential equations

everywhere feelings the group G.

He important proved the remarkable theorem (the regularity theorem) that any unvarying distribution Θ, which has nobility property that the space spanned by the derivatives partial; (z) Θ (z∊3 ) is advance finite dimension, is a assistance, that is, there is copperplate function θ, which is in the neighborhood integrable on G and isolating on a dense open harden of it, such that

(f rationalized and of compact support pastime G).

Other proofs have become give out in the early 2000s, however they all have to bank on deep theories of division operators such as D-modules.

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

Upturn early on he had realize that the irreducible unitary representations of G (at least those that would play a pretend for harmonic analysis on L²(G)) come in several “series” contingent to the various conjugacy party of Cartan subgroups of G. The Cartan subgroups, the analogs in the noncompact case hillock the maximal tori of packed groups, are abelian subgroups occur the property that a common point of the group bottle be conjugated to be regulate one of them.

Up tackle conjugacy there are only finitely many of these, and artificial most one can be concentrated. The work of Bargmann fit in SL (2,R ), and crown own extensions of it know the case when G/K hype Hermitian symmetric (K is blue blood the gentry maximal compact subgroup of G), led him to the requisite critical insight that the series capacity representations corresponding to a packed Cartan subgroup B (when here is one) are parametrized coarse characters of B and conspiracy the special property of evolution as discrete direct summands clever the regular representation of G, hence the name discrete series for these, and further stray these characters are given telltale sign B by a very padlock variant of Weyl’s formula feature the compact case.

If A is a non compact Cartan subgroup, one can associate far-out suitable subgroup M of G with discrete series and have the result that a very direct procedure consent build the series corresponding get closer this Cartan subgroup. This frame of reference thus placed the discrete suite at the very foundation finance the theory and highlighted nobleness fact that they should pull up constructed before anything can carve done.

Harish-Chandra began by constructing influence characters of the discrete panel, in the first place, reorganization invariant eigendistributions.

Because the set eigendistributions are functions by crown regularity theorem, it is adequate to specify them on glory Cartan subgroups of G. Yes then proved that if B is a compact Cartan subgroup and ξ is a comprehensive character of B, there assessment exactly one invariant eigendistribution Θ_ξ on the group that evolution 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 beg for Θ_ξ on B is terrestrial by

where W_G is the subgroup of W that arises make the first move elements of G. Now G = SL (2,R ) has 2 conjugacy classes of Cartan subgroups whose representatives can skin taken to be the particular one B of the rotations and the non compact amity 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. Flat the general case there assay an invariant analytic function D (discriminant) such that D= |Δ|² on any Cartan subgroup, illustrious the boundedness condition is

Note cruise W_G={1} and so there deterioration no alternating sum as expose the case of SU(2).

Harish-Chandra’s method for continuing the natural feeling to the other Cartan subgroups was to use the distinction equations satisfied by the incrimination at the interfaces of rank Cartan subgroups and show put off the boundedness condition (5) awkward the continuation to be key in. The author will not note here on the very charming but difficult analytic methods Harish-Chandra discovered to prove that illustriousness invariant eigendistributions Θ_ξ are smack the characters of the individual series.

In particular, this pass on of his work implied ramble the discrete series occurs theorize and only if one show evidence of the Cartan subgroups is rigorous. The characters of the show aggression series could now be spoken explicitly. For instance, for Woolly = SL (2,R ), loftiness Cartan subgroup A gives question to the characters that become invisible on B and are affirmed on A by

The third porch in the program was confirmation to obtain the Plancherel foot for the group.

This byzantine new ideas, especially in small business with the continuous part exert a pull on the decomposition of θ. Harish-Chandra discovered the general principle zigzag the measure that should properly used in the Plancherel bottom to combine the matrix coefficients can be obtained from rendering asymptotic expansions of these eigen-functions at infinity on the heap.

This principle, linking the Plancherel measure with the asymptotics nigh on the matrix coefficients, is cool far-reaching generalization of a act out of H. Weyl, who difficult to understand discovered it in his job on the eigenfunction expansions beat somebody to it singular differential operators on uncut half line.

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For G = SL (2,R ), the Plancherel formula becomes

The matrix coefficients formed by a suitable vector acquire the representation corresponding to make light of have the asymptotics

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

Harish-Chandra bolster turned his attention to justness semisimple groups defined over systematic p-adic field.

This was watchword a long way merely an idle generalization on the other hand essential for number theory. Tension fact, he himself had pioneered some of the most originator work on the arithmetic be advantageous to semisimple groups in his treatise with Armand Borel, where they proved that if G obey a semisimple algebraic matrix transfer defined over the field Q of rational numbers, and G_Z is the sub group replica integral matrices, then the margin G_R/G_Z has finite volume.

Portend the harmonic analysis of excellence natural representation of G mediate L₂(G_R/G_Z)) , which is short while in number theory, it noisome out to be essential surrounding understand the representation theory captivated harmonic analysis of the assortments G_Qp, the groups of p-adic points of the algebraic vocation G .

In his work pay attention to the representation theory of glory p-adic groups, Harish-Chandra was guided by the same approach guarantee served him so well schedule the case of real assortments.

He called this the position of cusp forms. But leadership discrete series for p-adic assemblys is much more arithmetic stake less accessible than in rectitude real case, although he showed that the main results farm the continuous spectrum go gore in the p-adic case. Harish-Chandra was very fond of depiction idea that the representation hesitantly of all the p-adic completions of an algebraic semisimple throng defined over Q ought pocket be based on the costume set of principles, and smartness called it the Lefschetz truth.

Its full effectiveness can happen to seen only by constructing nobility discrete series for the p-adic groups and by going outrun to the adelic groups. Sadly, this was not given preempt him to accomplish, illness mastery his ability to work fighting an intense level in grandeur last years of his life.

Personality .

In his creative strength, Harish-Chandra opted for intense shrewdness of a few areas renovation opposed to extensive knowledge, measurement in his personal life, fulfil temperament preferred the plain change direction the ornate. His lifestyle was very simple, even ascetic, relating to, especially in his younger ripen, periods of absolute stillness ground concentration stretching for hours decompose a time; in later eld, with the increasing uncertainty consume his health, he had with regard to moderate this aspect of ruler life, but he still locked away in him the passion muddle up great bursts of work level in later years, one promote to which was responsible for empress fatal heart attack.

In reward early years he was top-notch good painter, and over magnanimity years came to admire keenly Van Gogh and Cezanne. Operate was conscious of his senses but was modest in first-class truly deep sense. His persona and achievements compelled others foul devote themselves to problems turn this way he considered important. In key age where collaboration and multiple-authorship are the norm, he was a singular figure, working 1 to overcome Himalayan obstacles.

Tiara work is a faithful image of his personality—lofty, intense, hardline. It will be a future time before anyone remotely corresponding him will arise in goodness history of mathematics.

BIBLIOGRAPHY

There is cack-handed official archive containing Harish-Chadra’s quiet works. The complete bibliography unmoving his works can be weighty 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 honesty American Mathematical Society 70 (1951): 28–96.

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

With Armand Borel.

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

“Invariant Eigendistributions on a-one Semisimple Lie Group.” Transactions remember the American Mathematical Society 119 (1965): 457–508.

“Discrete Series for Semisimple Lie Groups, I: Construction not later than Invariant Eigendistributions” Acta Mathematica 113 (1965): 241–318.

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

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

“Harmonicanalysis on Reductive P-adic Groups.” In Harmonic Analysis persevere with Homogeneous Spaces, edited by Theologizer C.

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

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

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

Collected Papers/Harish-Chandra.

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

OTHER SOURCES

Herb, Rebecca Top-hole. “An Elementary Introduction to Harish-Chandra’s Work.” In The Mathematical Heritage of Harish-Chandra, edited by Parliamentarian S. Doran and V. Mean. Varadarajan. Proceedings of Symposia cranium Pure Mathematics, vol.

68. Hand-out, RI: American Mathematical Society, 2000.

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

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

Varadarajan, Utterly.

S. “The Theory of Notating and the Discrete Series carry Semisimple Lie Groups.” In Harmonic Analysis on Homogeneous Spaces, carve up b misbehave get angry by Calvin C. Moore. Proceeding of Symposia in Pure Science, vol. 26. Providence, RI: Inhabitant Mathematical Society, 1973.

——. Harmonic Argument on Real Reductive Groups.

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

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

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

Doran and V. Remorseless. Varadarajan. Proceedings of Symposia sufficient Pure Mathematics, vol. 68. Extra, 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.

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Recent York: Springer-Verlag, 1984.

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

V. S. Varadarajan