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Discrete Series Representation
In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group ''G'' that is a subrepresentation of the left regular representation of ''G'' on L²(''G''). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation. Properties If ''G'' is unimodular group, unimodular, an irreducible unitary representation ρ of ''G'' is in the discrete series if and only if one (and hence all) matrix coefficient :\langle \rho(g)\cdot v, w \rangle \, with ''v'', ''w'' non-zero vectors is square-integrable on ''G'', with respect to Haar measure. When ''G'' is unimodular, the discrete series representation has a formal dimension ''d'', with the property that :d\int \langle \rho(g)\cdot v, w \rangle \overlinedg =\langle v, x \rangle\overline for ''v'', ''w'', ''x'', ''y'' in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weight Lattice
In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n \times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomological Parabolic Induction
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related. Notation and terminology *''G'' is a connected reductive real affine algebraic group (for simplicity; the theory works for more general groups), and ''g'' is the Lie algebra of ''G''. *''K'' is a maximal compact subgroup of ''G''. *A (g,K)-module is a vector space with compatible actions of ''g'' and ''K'', on which the action of ''K'' is ''K''-finite. A representation of ''K'' is called K-finite if every vector is contained in a finite-dimensional representation of ''K''. *''W''''K'' is the subspace of ''K''-finite vectors of a representation ''W'' of ''K''. *R(''g'',''K'') is the Hecke algebra of ''G'' of all distributions on ''G'' with support in ''K'' that are left and right ''K'' finite. This is a ring which does not have an identity but has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Spinor
In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Hamilton and in 1928 by Paul Dirac. The question which concerned Dirac was to factorise formally the Laplace operator of the Minkowski space, to get an equation for the wave function which would be compatible with special relativity. Formal definition In general, let ''D'' be a first-order differential operator acting on a vector bundle ''V'' over a Riemannian manifold ''M''. If : D^2=\Delta, \, where ∆ is the (positive, or geometric) Laplacian of ''V'', then ''D'' is called a Dirac operator. Note that there are two different conventions as to how the Laplace operator is defined: the "analytic" Laplacian, which could be characterized in \R^n as \Delta=\nabla^2=\sum_^n\Big(\frac\Big)^2 (which is negative-definite, in the sense that \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Theorem
Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ''Halo'' video game series Periodicals and news portals * ''Index Magazine'', a publication for art and culture * Index.hr, a Croatian online newspaper * index.hu, a Hungarian-language news and community portal * ''The Index'' (Kalamazoo College), a student newspaper * ''The Index'', an 1860s European propaganda journal created by Henry Hotze to support the Confederate States of America * ''Truman State University Index'', a student newspaper Other arts, entertainment and media * The Index (band) * ''Indexed'', a Web cartoon by Jessica Hagy * ''Index'', album by Ana Mena Business enterprises and events * Index (retailer), a former UK catalogue retailer * INDEX, a market research fair in Lucknow, India * Index Corporation, a Japanese v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L2 Cohomology
L, or l, is the twelfth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''el'' (pronounced ), plural ''els''. History Lamedh may have come from a pictogram of an ox goad or cattle prod. Some have suggested that it represents a shepherd's staff. Typographic variants In most sans-serif typefaces, the lowercase letter ''ell'' , written as the glyph , may be difficult to distinguish from the uppercase letter "eye" (written as the glyph ); in some serif typefaces, the glyph may be confused with the glyph , the digit ''one''. To avoid such confusion, some newer computer fonts (such as Trebuchet MS) have a finial, a curve to the right at the bottom of the lowercase letter ''ell''. Other style variants are provided in script typefaces and display typefaces. All these variants of the letter are encoded in Unicode as or , allowing presentation to be chosen accor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tempered Representation
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L''p'' space :''L''2+ε(''G'') for any ε > 0. Formulation This condition, as just given, is slightly weaker than the condition that the matrix coefficients are square-integrable, in other words lie in :''L''2(''G''), which would be the definition of a discrete series representation. If ''G'' is a linear semisimple Lie group with a maximal compact subgroup ''K'', an admissible representation ρ of ''G'' is tempered if the above condition holds for the ''K''-finite matrix coefficients of ρ. The definition above is also used for more general groups, such as ''p''-adic Lie groups and finite central extensions of semisimple real algebraic groups. The definition of "tempered representation" makes sense for arbitrary unimodular locally compact groups, but on groups with infinite centers such as infinite central extensio ... [...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'' ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz Distribution
Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than Solution of a differential equation, classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function, Dirac delta function. A Function (mathematics), function f is normally thought of as on the in the function Domain (function), domain by "sending" a point x in the domain t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl 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 algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra Correspondence
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] |
