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Harish-Chandra
Harish-Chandra Fellow of the Royal Society, FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups. Early life Harish-Chandra was born in Kanpur. He was educated at BNSD Inter College, B.N.S.D. College, Kanpur and at the University of Allahabad. After receiving his master's degree in Physics in 1943, he moved to the Indian Institute of Science, Bangalore for further studies under Homi J. Bhabha. In 1945, he moved to University of Cambridge, and worked as a research student under Paul Dirac. While at Cambridge, he attended lectures by Wolfgang Pauli, and during one of them pointed out a mistake in Pauli's work. The two were to become lifelong friends. During this time he became increasingly interested in mathematics. At Cambridge he obtained his PhD in 1947. Honors and awards He was a member of the United States National Academy of Scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's Regularity Theorem
In mathematics, Harish-Chandra's regularity theorem, introduced by , states that every invariant eigendistribution on a semisimple Lie group, and in particular every character of an irreducible unitary representation on a Hilbert space, is given by a locally integrable function. proved a similar theorem for semisimple ''p''-adic groups. had previously shown that any invariant eigendistribution is analytic on the regular elements of the group, by showing that on these elements it is a solution of an elliptic differential equation. The problem is that it may have singularities on the singular elements of the group; the regularity theorem implies that these singularities are not too severe. Statement A distribution on a group ''G'' or its Lie algebra is called invariant if it is invariant under conjugation by ''G''. A distribution on a group ''G'' or its Lie algebra is called an eigendistribution if it is an eigenvector of the center of the universal enveloping algebra of ''G'' (id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's C-function
In mathematics, Harish-Chandra's ''c''-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and introduced a more general ''c''-function called Harish-Chandra's (generalized) ''C''-function. introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's ''c''-function. Gindikin–Karpelevich formula The c-function has a generalization ''c''''w''(λ) depending on an element ''w'' of the Weyl group. The unique element of greatest length ''s''0, is the unique element that carries the Weyl chamber \mathfrak_+^* onto -\mathfrak_+^*. By Harish-Chandra's integral formula, ''c''''s''0 is Harish-Chandra's c-function: : c(\lambda)=c_(\lambda). The c-functions are in general defined by the equation : \displaystyle A(s,\lamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra Isomorphism
In mathematics, the Harish-Chandra isomorphism, introduced by , is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center \mathcal(U(\mathfrak)) of the universal enveloping algebra U(\mathfrak) of a reductive Lie algebra \mathfrak to the elements S(\mathfrak)^W of the symmetric algebra S(\mathfrak) of a Cartan subalgebra \mathfrak that are invariant under the Weyl group W. Introduction and setting Let \mathfrak be a semisimple Lie algebra, \mathfrak its Cartan subalgebra and \lambda, \mu \in \mathfrak^* be two elements of the weight space (where \mathfrak^* is the dual of \mathfrak) and assume that a set of positive roots \Phi_+ have been fixed. Let V_\lambda and V_\mu be highest weight modules with highest weights \lambda and \mu respectively. Central characters The \mathfrak-modules V_\lambda and V_\mu are representations of the universal enveloping algebra U(\mathfrak) and its center acts on the modules by scalar m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra Homomorphism
In mathematical representation theory, a Harish-Chandra homomorphism is a homomorphism from a subalgebra of the universal enveloping algebra of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra. A particularly important special case is the Harish-Chandra isomorphism identifying the center of the universal enveloping algebra with the invariant polynomials on a Cartan subalgebra. In the case of the ''K''-invariant elements of the universal enveloping algebra for a maximal compact subgroup ''K'', the Harish-Chandra homomorphism was studied by . References * *{{Citation , last1=Howe , first1=Roger E. , editor1-last=Doran , editor1-first=Robert S. , editor2-last=Varadarajan. , editor2-first=V. S. , title=The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) , url=https://books.google.com/books?id=mk-4pl9IftMC&pg=321 , publisher=American Mathematical Society The American Mathematical Society (AMS) is an association of professional mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra Module
In mathematics, specifically in the representation theory of Lie groups, a Harish-Chandra module, named after the Indian mathematician and physicist Harish-Chandra, is a representation of a real Lie group, associated to a general representation, with regularity and finiteness conditions. When the associated representation is a (\mathfrak,K)-module, then its Harish-Chandra module is a representation with desirable factorization properties. Definition Let ''G'' be a Lie group and ''K'' a compact subgroup of ''G''. If (\pi,V) is a representation of ''G'', then the ''Harish-Chandra module'' of \pi is the subspace ''X'' of ''V'' consisting of the K-finite smooth vectors in ''V''. This means that ''X'' includes exactly those vectors ''v'' such that the map \varphi_v : G \longrightarrow V via :\varphi_v(g) = \pi(g)v is smooth, and the subspace :\text\ is finite-dimensional. Notes In 1973, Lepowsky showed that any irreducible (\mathfrak,K)-module ''X'' is isomorphic to the Harish-Cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's Ξ Function
In mathematical harmonic analysis, Harish-Chandra's ''Ξ'' function is a special spherical function on a semisimple Lie group, studied by . Harish-Chandra used it to define Harish-Chandra's Schwartz space. gives a detailed description of the properties of ''Ξ''. Definition :\Xi(g)=\int_Ka(kg)^\rho dk, where *''K'' is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition ''G''=''NAK'' *''g'' is an element of ''G'' *''ρ'' is a Weyl vector In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the c ... *''a''(''g'') is the element ''a'' in the Iwasawa decomposition ''g''=''nak'' References * * {{DEFAULTSORT:Harish-Chandra's Xi function Harmonic analysis Representation theory Special functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's Schwartz Space
In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by . It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group. Definition The definition of the Schwartz space uses Harish-Chandra's Ξ function and his ''σ'' function. The ''σ'' function is defined by : \sigma(x)=\, X\, for ''x''=''k'' exp ''X'' with ''k'' in ''K'' and ''X'' in ''p'' for a Cartan decomposition ''G'' = ''K'' exp ''p'' of the Lie group ''G'', where , , ''X'', , is a ''K''-invariant Euclidean norm on ''p'', usually chosen to be the Killing form. . The Schwartz space on ''G'' consists roughly of the functions all of whose derivatives are rapidly decreasing compared to ''Ξ''. More precisely, if ''G'' is connected then the Schwartz space consists of all smooth funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra Transform
In mathematical representation theory, the Harish-Chandra transform is a linear map from functions on a reductive Lie group to functions on a parabolic subgroup. It was introduced by . The Harish-Chandra transform ''f''''P'' of a function ''f'' on the group ''G'' is given by : f^P(m) =a^\int_Nf(nm)\,dn where ''P'' = ''MAN'' is the Langlands decomposition of a parabolic subgroup. References * *{{Citation , last1=Wallach , first1=Nolan R , title=Real reductive groups. I , publisher=Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes referen ... , location=Boston, MA , series=Pure and Applied Mathematics , isbn=978-0-12-732960-4 , mr=929683 , year=1988 , volume=132 , url-access=registration , url=https://archive.org/details/realreductivegro0000wall Represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's Character Formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula. By definition, the character \chi of a representation \pi of ''G'' is the trace of \pi(g), as a function of a group element g\in G. The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory.. The prize is named after Frank Nelson Cole, who served the Society for 25 years. The Cole Prize in algebra was funded by Cole himself, from funds given to him as a retirement gift; the prize fund was later augmented by his son, leading to the double award.. To be eligible for the Cole prize, the author must be a member of the American Mathematical Society or the paper should appear in a recognized North American journal. The first award for algebra was made in 1928 to L. E. Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indian Institute Of Science
The Indian Institute of Science (IISc) is a public, deemed, research university for higher education and research in science, engineering, design, and management. It is located in Bengaluru, in the Indian state of Karnataka. The institute was established in 1909 with active support from Jamsetji Tata and thus is also locally known as the ''"Tata Institute"''. It is ranked among the most prestigious academic institutions in India and has the highest citation per faculty among all the universities in the world. It was granted the deemed to be university status in 1958 and the Institute of Eminence status in 2018. History After an accidental meeting between Jamsetji Tata and Swami Vivekananda, on a ship in 1893 where they discussed Tata's plan of bringing the steel industry to India, Tata wrote to Vivekananda five years later: "I trust, you remember me as a fellow-traveller on your voyage from Japan to Chicago. I very much recall at this moment your views on the growth of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
