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Howe Duality Conjecture
In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorphic representations over a global field. The theta correspondence was introduced by Roger Howe in . Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in . The Shimura correspondence as constructed by Jean-Loup Waldspurger in and may be viewed as an instance of the theta correspondence. Statement Setup Let F be a local or a global field, not of characteristic 2. Let W be a symplectic vector space over F, and Sp(W) the symplectic group. Fix a reductive dual pair (G,H) in Sp(W). There is a classification of reductive dual pairs. Local theta correspondence F is now a local field. Fix a non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character (mathematics)
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. Multiplicative character A multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the unit circle); other such homomorphisms are then called ''quasi-characters''. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent, i.e. if \chi_1,\chi_2, \ldots , \chi_n are different cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...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] |
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] |
Reductive Dual Pair
In the mathematical field of representation theory, a reductive dual pair is a pair of subgroups (''G'', ''G''′) of the isometry group Sp(''W'') of a symplectic vector space ''W'', such that ''G'' is the centralizer of ''G''′ in Sp(''W'') and vice versa, and these groups act reductively on ''W''. Somewhat more loosely, one speaks of a dual pair whenever two groups are the mutual centralizers in a larger group, which is frequently a general linear group. The concept was introduced by Roger Howe in . Its strong ties with Classical Invariant Theory are discussed in . Examples * The full symplectic group ''G'' = Sp(''W'') and the two-element group ''G''′, the center of Sp(''W''), form a reductive dual pair. The double centralizer property is clear from the way these groups were defined: the centralizer of the group ''G'' in ''G'' is its center, and the centralizer of the center of any group is the group itself. The group ''G''′, consists of the identity transformat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sun Binyong
Sun Binyong (; born November 1976) is a Chinese mathematician. He is an academician of the Chinese Academy of Sciences (CAS). Early life and education Sun was born in Putuo District, Zhoushan, Zhejiang in November 1976, the second of three sons. His mother Liu Yadi () is a housewife. His father Sun Kaizhu () was a carpenter. He attended Shuangtang Middle School, Putuo Middle School and the High School attached to Tsinghua University. After high school, he entered Zhejiang University, where he graduated in 1999. In December 2004 he earned his doctorate degree from Hong Kong University of Science and Technology under the supervision of Li Jianshu. Career From January to September 2005 he was a postdoctoral researcher at the Swiss Federal Institute of Technology Zurich in Switzerland. In September 2005 he became research associate at the Institute of Mathematics and Systems Sciences, Chinese Academy of Sciences (CAS), and was promoted to researcher in 2011. Personal life Sun is m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wee Teck Gan
Gan Wee Teck (; born 11 March 1972) is a Malaysian mathematician. He is a Distinguished Professor of Mathematics at the National University of Singapore (NUS). He is known for his work on automorphic forms and representation theory in the context of the Langlands program, especially the theory of theta correspondence, the Gan–Gross–Prasad conjecture and the Langlands program for Brylinski–Deligne covering groups. Biography Though born in Malaysia, Gan grew up in Singapore and attended Pei Hwa Presbyterian Primary School, the Chinese High School, and Hwa Chong Junior College. He did his undergraduate studies at Churchill College, Cambridge University, followed by graduate studies at Harvard University, working under Benedict Gross and obtaining his Ph.D. in 1998. He was subsequently a faculty member at Princeton University (1998–2003) and University of California, San Diego (2003–2010) before moving to the National University of Singapore in 2010. Contributions With his ... [...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] |
Archimedean Property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two positive numbers ''x'' and ''y'', there is an integer ''n'' such that ''nx'' > ''y''. It also means that the set of natural numbers is not bounded above. Roughly speaking, it is the property of having no ''infinitely large'' or ''infinitely small'' elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ ''On the Sphere and Cylinder''. The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields. An algebraic structure in which any two non-zero elements are ''comparabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuspidal Representation
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term ''cuspidal'' is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups. When the group is the general linear group \operatorname_2, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform ( newform) corresponds to a cuspidal representation. Formulation Let ''G'' be a reductive algebraic group over a number field ''K'' and let A denote the adeles of ''K''. The group ''G''(''K'') embeds diagonally in the group ''G''(A) by sending ''g'' in ''G''(''K'') to the tuple (''g''''p'')''p'' in ''G''(A) with ''g'' = ''g''''p'' for all (finite and infinite) primes ''p''. Let ''Z'' d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Rallis
Stephen James Rallis (May 17, 1942 – April 17, 2012) was an American mathematician who worked on group representations, automorphic forms, the Siegel–Weil formula, and Langlands L-functions. Career Rallis received a B.A. in 1964 from Harvard University, a Ph.D. in 1968 from the Massachusetts Institute of Technology, and spent 1968–1970 at the Institute for Advanced Study in Princeton. After two years at Stony Brook, two years at Universite de Strasbourg, and several visiting positions, he joined the faculty at Ohio State University in 1977 and stayed there for the rest of his career. Work Beginning in the 1970s, Rallis and Gérard Schiffmann wrote a series of papers on the Weil representation. This led to Rallis's work with Kudla in which they developed a far-reaching generalization of the Siegel–Weil formula: the regularized Siegel–Weil formula and the first term identity. These results have prompted other mathematicians to extend Siegel–Weil to other cases. Ralli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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