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Geometric Satake Equivalence
In mathematics, the Satake isomorphism, introduced by , identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by . Statement Classical Satake isomorphism. Let G be a semisimple algebraic group, K be a non-Archimedean local field and O be its ring of integers. It's easy to see that Gr = G(K)/G(O) is grassmannian. For simplicity, we can think that K = \Z/p\Z((x)) and O = \Z/p\Z x , p a prime number; in this case, Gr is a infinite dimensional algebraic variety . One denotes the category of all compactly supported spherical functions on G(K) biinvariant under the action of G(O) as \Complex_c (O) \backslash G(K)/G(O), \Complex the field of complex numbers, which is a Hecke algebra and can be also treated as a group scheme over \Complex . Let T(\Complex) be the maximal torus of G(\Complex) , W be the Weyl group of G . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra Of A Locally Compact Group
In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution. Definition Let (''G'',''K'') be a pair consisting of a unimodular locally compact topological group ''G'' and a closed subgroup ''K'' of ''G''. Then the space of bi-''K''-invariant continuous functions of compact support :''C'' 'K''\''G''/''K'' can be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted :''H''(''G''//''K'') and called the Hecke ring of the pair (''G'',''K''). If we start with a Gelfand pair then the resulting algebra turns out to be commutative. Examples SL(2) In particular, this holds when :''G'' = ''SL''''n''(''Q''''p'') and ''K'' = ''SL''''n''(''Z''''p'') and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald. GL(2) On the other hand, in the case :''G'' = ''GL''2(Q) and ''K'' = ''GL''2(Z) we have the classical Hecke alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perverse Sheaf
The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular. The concept was introduced in the work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a consequence of the Riemann-Hilbert correspondence, which establishes a connection between the derived categories regular holonomic D-modules and constructible sheaves. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more general complexes thereof); a perverse sheaf ''is'' in general represented by a complex of sheaves. The concept of perverse sheaves is already implicit in a 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules. A key observation was that the intersection homology of Mark Goresky and Robert MacPherson could be described using sheaf complexes that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tannakian Formalism
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to generalise the category of linear representations of an algebraic group ''G'' defined over ''K''. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory. The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups ''G'' and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups ''G'' which are profinite groups. The gist of the theory is that the fiber functor Φ of the Galois the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perverse Sheaf
The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces (intersection homology of Mark Goresky and Robert MacPherson) and the algebraic theory of differential equations (microlocal calculus and holonomic D-modules of Joseph Bernstein, Masaki Kashiwara and Takahiro Kawai). It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Dual
In representation theory, a branch of mathematics, the Langlands dual ''L''''G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a field ''k'', then ''L''''G'' is an extension of the absolute Galois group of ''k'' by a complex Lie group. There is also a variation called the Weil form of the ''L''-group, where the Galois group is replaced by a Weil group. Here, the letter ''L'' in the name also indicates the connection with the theory of L-functions, particularly the ''automorphic'' L-functions. The Langlands dual was introduced by in a letter to A. Weil. The ''L''-group is used heavily in the Langlands conjectures of Robert Langlands. It is used to make precise statements from ideas that automorphic forms are in a sense functorial in the group ''G'', when ''k'' is a global field. It is not exactly ''G'' with respect to which automorphic forms and representations are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic image of will also contain a homomorphic image of the Grothendieck group of . The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation. Grothendieck group of a commutative monoid Motivation Given a commutative monoid , "the most general" abelian group that arises from is to be constructed by introducing inverse elements to all elements of . Such an abelian group always exists; it is called the Grothendieck group of . It is character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The Category (mathematics), category of group schemes is somewhat better behaved than that of Group variety, group varieties, since all homomorphisms have Kernel (category theory), kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The ini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra Of A Locally Compact Group
In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution. Definition Let (''G'',''K'') be a pair consisting of a unimodular locally compact topological group ''G'' and a closed subgroup ''K'' of ''G''. Then the space of bi-''K''-invariant continuous functions of compact support :''C'' 'K''\''G''/''K'' can be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted :''H''(''G''//''K'') and called the Hecke ring of the pair (''G'',''K''). If we start with a Gelfand pair then the resulting algebra turns out to be commutative. Examples SL(2) In particular, this holds when :''G'' = ''SL''''n''(''Q''''p'') and ''K'' = ''SL''''n''(''Z''''p'') and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald. GL(2) On the other hand, in the case :''G'' = ''GL''2(Q) and ''K'' = ''GL''2(Z) we have the classical Hecke algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Functions
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
