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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] |
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] |
Weight (representation Theory)
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 are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compositio Mathematica
''Compositio Mathematica'' is a monthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935. It is owned by the Foundation Compositio Mathematica, and since 2004 it has been published on behalf of the Foundation by the London Mathematical Society in partnership with Cambridge University Press. According to the ''Journal Citation Reports'', the journal has a 2020 2-year impact factor of 1.456 and a 2020 5-year impact factor of 1.696. The editors-in-chief are Jochen Heinloth, Bruno Klingler, Lenny Taelman, and Éric Vasserot. Early history The journal was established by L. E. J. Brouwer in response to his dismissal from ''Mathematische Annalen'' in 1928. An announcement of the new journal was made in a 1934 issue of the ''American Mathematical Monthly''. In 1940 the publication of the journal was suspended due to the German occupation of the Netherlands Despite Dutch neutrality, Nazi Germany invaded the Netherlands on 10 May 1940 as part of Fall Gelb (Case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a d ... [...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 extensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Loup Waldspurger
Jean-Loup Waldspurger (born July 2, 1953) is a French mathematician working on the Langlands program and related areas. He proved Waldspurger's theorem, the Waldspurger formula, and the local Gan–Gross–Prasad conjecture for orthogonal groups. He played a role in the proof of the fundamental lemma, reducing the conjecture to a version for Lie algebras. This formulation was ultimately proven by Ngô Bảo Châu. Education Waldspurger attained his doctorate at École normale supérieure in 1980, under supervision of Marie-France Vignéras. Scientific work J.-L. Waldspurger's work concerns the theory of automorphic forms. He highlighted the links between Fourier coefficients of modular shapes of half full weight and function values L or periods of modular shapes of full weight. With C. Moeglin, he demonstrated Jacquet's conjecture describing the discrete spectrum of the GL(n) groups. Other works are devoted to orbital integrals on p-adic groups: unipotent orbital integrals, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Langlands Conjectures
In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ''G'' over a local field ''F'', and representations of the Langlands group of ''F'' into the L-group of ''G''. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups. Local Langlands conjectures for GL1 The local Langlands conjectures for GL1(''K'') follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL1(''K'')= ''K''* to the abelianization of the Weil group. In particular irreducible smooth representations of GL1(''K'') are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL1(C). This gives the Langlands corresponden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands–Deligne Local Constant
In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor or local Artin root number (up to an elementary real function of ''s''), is an elementary function associated with a representation of the Weil group of a local field. The functional equation :L(ρ,''s'') = ε(ρ,''s'')L(ρ∨,1−''s'') of an Artin L-function has an elementary function ε(ρ,''s'') appearing in it, equal to a constant called the Artin root number times an elementary real function of ''s'', and Langlands discovered that ε(ρ,''s'') can be written in a canonical way as a product :ε(ρ,''s'') = Π ε(ρ''v'', ''s'', ψ''v'') of local constants ε(ρ''v'', ''s'', ψ''v'') associated to primes ''v''. Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis. proved the existence of the local constant ε(ρ''v'', ''s'', ψ''v'') up to sign. The original proof of the existence of the local constants by used local methods and was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Langlands Conjectures
In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ''G'' over a local field ''F'', and representations of the Langlands group of ''F'' into the L-group of ''G''. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups. Local Langlands conjectures for GL1 The local Langlands conjectures for GL1(''K'') follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL1(''K'')= ''K''* to the abelianization of the Weil group. In particular irreducible smooth representations of GL1(''K'') are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL1(C). This gives the Langlands corresponden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metaplectic Group
In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence. Definition The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2''n''(R) and called the metaplectic group. The metaplectic group Mp2(R) is ''not'' a matrix group: it has no faithful finite-dimensional representations. Therefore, the question of its explicit realization is nontrivial. It has faithful ir ... [...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] |
Diagonal Subgroup
In the mathematical discipline of group theory, for a given group the diagonal subgroup of the ''n''-fold direct product is the subgroup :\. This subgroup is isomorphic to Properties and applications * If acts on a set the ''n''-fold diagonal subgroup has a natural action on the Cartesian product induced by the action of on defined by :(x_1, \dots, x_n) \cdot (g, \dots, g) = (x_1 \!\cdot g, \dots, x_n \!\cdot g). * If acts - transitively on then the -fold diagonal subgroup acts transitively on More generally, for an integer if acts -transitively on acts -transitively on * Burnside's lemma Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, the orbit-counting theorem, or the Lemma that is not Burnside's, is a result in group theory that is often useful in taking account of symmetry when ... can be proved using the action of the twofold diagonal subgroup. See also * Diagonalizable group References *. Group theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
