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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 extensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Group
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of Characteristic (algebra), characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable Lie algebra, solvable ideals; * the Radical of a Lie algebra, radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Decomposition
In mathematics, the Langlands decomposition writes a parabolic subgroup ''P'' of a semisimple Lie group as a product P=MAN of a reductive subgroup ''M'', an abelian subgroup ''A'', and a nilpotent subgroup ''N''. Applications A key application is in parabolic induction, which leads to the Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...: if G is a reductive algebraic group and P=MAN is the Langlands decomposition of a parabolic subgroup ''P'', then parabolic induction consists of taking a representation of MA, extending it to P by letting N act trivially, and inducing the result from P to G. See also * Lie group decompositions References Sources * A. W. Knapp, Structure theory of semisimple Lie groups. . Lie groups Algebraic groups {{Mathan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Representation
In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ( Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of ''V'' to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complementary Series Representation
In mathematics, complementary series representations of a reductive real or ''p''-adic Lie groups are certain irreducible unitary representations that are not tempered and do not appear in the decomposition of the regular representation into irreducible representations. They are rather mysterious: they do not turn up very often, and seem to exist by accident. They were sometimes overlooked, in fact, in some earlier claims to have classified the irreducible unitary representations of certain groups. Several conjectures in mathematics, such as the Selberg conjecture, are equivalent to saying that certain representations are not complementary. For examples see the representation theory of SL2(R). Elias M. Stein Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, w ... (1972) constructed so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Integral
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of 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, 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 the representation. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Series Representation
In mathematics, the principal series representations of certain kinds of topological group ''G'' occur in the case where ''G'' is not a compact group. There, by analogy with spectral theory, one expects that the regular representation of ''G'' will decompose according to some kind of continuous spectrum, of representations involving a continuous parameter, as well as a discrete spectrum. The ''principal series'' representations are some induced representations constructed in a uniform way, in order to fill out the continuous part of the spectrum. In more detail, the unitary dual is the space of all representations relevant to decomposing the regular representation. The discrete series consists of 'atoms' of the unitary dual (points carrying a Plancherel measure > 0). In the earliest examples studied, the rest (or most) of the unitary dual could be parametrised by starting with a subgroup ''H'' of ''G'', simpler but not compact, and building up induced representations using represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SL2(R)
In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: : \mbox(2,\mathbf) = \left\. It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics. SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More specifically, :PSL(2, R) = SL(2, R) / , where ''I'' denotes the 2 × 2 identity matrix. It contains the modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL±(2, R), the group of real 2 × 2 matrices with determinant ±1; this is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Series
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, 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 the representation. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics." The Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. To oversimplify, the fundamental lemma of the project posits a direct connection between the generalized fundamental representation of a finite field with its group extension to the automorphic forms under which it is invariant. This is accomplished through abstraction to higher dimensional integrati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
