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Multiplicity-one Theorem
In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question is the number of times a given abstract group representation is realised in a certain space, of square-integrable functions, given in a concrete way. A multiplicity one theorem may also refer to a result about the restriction of a representation of a group ''G'' to a subgroup ''H''. In that context, the pair (''G'', ''H'') is called a strong Gelfand pair. Definition Let ''G'' be a reductive algebraic group over a number field ''K'' and let A denote the adeles of ''K''. Let ''Z'' denote the centre of ''G'' and let be a continuous unitary character from ''Z''(''K'')\Z(A)× to C×. Let ''L''20(''G''(''K'')/''G''(A), ) denote the space of cusp forms with central character ω on ''G''(A). This space decomposes into a direct sum of Hilbert spaces :L^2_0(G(K) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Representation
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ''G''(A''F''), for an algebraic group ''G'' and an algebraic number field ''F'', is a complex-valued function on ''G''(A''F'') that is le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum Of Hilbert Spaces
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules. Construction for vector spaces and abelian groups We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gan–Gross–Prasad Conjecture
In mathematics, the Gan–Gross–Prasad conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The problem originated from a conjecture of Gross and Prasad for special orthogonal groups but was later generalized to include all four classical groups. In the cases considered, it is known that the multiplicity of the restrictions is at most one and the conjecture describes when the multiplicity is precisely one. Motivation A motivating example is the following classical branching problem in the theory of compact Lie groups. Let \pi be an irreducible finite dimensional representation of the compact unitary group U(n), and consider its restriction to the naturally embedded subgroup U(n-1). It is known that this restriction is multiplicity-free, but one may ask precisely which irreducible representations of U(n-1) occur in the restriction. By the Cartan–Weyl theory of highest weig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Linear Group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant :\det\colon \operatorname(n, F) \to F^\times. where ''F''× is the multiplicative group of ''F'' (that is, ''F'' excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When ''F'' is a finite field of order ''q'', the notation is sometimes used. Geometric interpretation The special linear group can be characterized as the group of ''volume and orientation preserving'' linear transformations of R''n''; this corresponds to the interpretation of the determinant as measuring change in volume and orientation. Lie subgroup When ''F'' is R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whittaker Model
In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as ''GL''2 over a finite or local or global field on a space of functions on the group. It is named after E. T. Whittaker even though he never worked in this area, because pointed out that for the group SL2(R) some of the functions involved in the representation are Whittaker functions. Irreducible representations without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation ''θ''10 of the symplectic group Sp4 is the simplest example of a degenerate representation. Whittaker models for GL2 If ''G'' is the algebraic group ''GL''2 and F is a local field, and is a fixed non-trivial character of the additive group of F and is an irreducible representation of a general linear group ''G''(F), then the Whittaker model for is a representation on a space of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Linear Group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of invertible matrices of real numbers, and is denoted by GL''n''(R) or . More generally, the general linear group of degree ''n'' over any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cusp Form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. Introduction A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient ''a''0 in the Fourier series expansion (see ''q''-expansion) :\sum a_n q^n. This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane via the transformation :z\mapsto z+1. For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter. In all cases, though, the limit as ''q'' → 0 is the limit in the upper half-plane as the imaginary part of ''z'' → ∞. Taking the quotient by the modular group, this limit corresponds to a cusp of a modular curve (in the sense of a point added for compactification). So, the definition amounts to saying that a cusp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Admissible Representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. Real or complex reductive Lie groups Let ''G'' be a connected reductive (real or complex) Lie group. Let ''K'' be a maximal compact subgroup. A continuous representation (π, ''V'') of ''G'' on a complex Hilbert space ''V''I.e. a homomorphism (where GL(''V'') is the group of bounded linear operators on ''V'' whose inverse is also bounded and linear) such that the associated map is continuous. is called admissible if π restricted to ''K'' is unitary and each irreducible unitary representation of ''K'' occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of ''G''. An admissible representation π induces a (\mathfrak,K)-module which is easier to deal with as it is an algebraic object. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. Real or complex reductive Lie groups Let ''G'' be a connected reductive (real or complex) Lie group. Let ''K'' be a maximal compact subgroup. A continuous representation (π, ''V'') of ''G'' on a complex Hilbert space ''V''I.e. a homomorphism (where GL(''V'') is the group of bounded linear operators on ''V'' whose inverse is also bounded and linear) such that the associated map is continuous. is called admissible if π restricted to ''K'' is unitary and each irreducible unitary representation of ''K'' occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of ''G''. An admissible representation π induces a (\mathfrak,K)-module which is easier to deal with as it is an algebraic obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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