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Topic: Automorphic representation

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	Automorphic form - Wikipedia, the free encyclopedia
		In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms.
		The subsequent notion of automorphic representation has proved of great technical value for dealing with G an algebraic group, treated as an adelic algebraic group.
		Under Poincaré's definition, an automorphic function is one which is analytic in its domain and is invariant under a denumerable infinite group of linear fractional transformations.
	en.wikipedia.org /wiki/Automorphic_representation (1017 words)

	Langlands program - Wikipedia, the free encyclopedia
		The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series (that is, the analogues of the Riemann zeta function constructed from Hecke characters).
		Langlands attached L-functions to these automorphic representations, and conjectured that every Artin L-function arising from a finite-dimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation.
		It is of the nature of an induced representation construction — what in the more traditional theory of automorphic forms had been called a 'lifting', known in special cases, and so is covariant (where a restricted representation is contravariant).
	www.wikipedia.org /wiki/Langlands_program (889 words)

	Automorphic representation (Site not responding. Last check: 2007-10-31)
		In mathematics, the general notion of automorphic form isthe extension to analytic functions, perhaps of several complex variables, of the theory of modular forms.
		The formulationrequires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology.
		One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is naturalfrom the point of view of functional analysis, though not soobviously for the number theory.
	www.therfcc.org /automorphic-representation-79318.html (671 words)

	Representations of Lie groups (Site not responding. Last check: 2007-10-31)
		A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebra s (indeed in the physics literature the distinction is often elided).
		Formally, a representation of a Lie group ''G on a vector space V (over a field K) is a group homomorphism G →Aut(V) from G to the automorphism group of V.
		A unitary representation is defined in the same way, except that G maps to unitary matrices ; the Lie algebra will then map to skew-hermitian matrices.
	www.serebella.com /encyclopedia/article-Representations_of_Lie_groups.html (961 words)

	Fields Institute - James Arthur
		Representation theory is the study of the deeper aspects of symmetry.
		Representation theory probes the hidden mathematical properties of symmetry in much the same way that spectroscopy analyzes hidden physical properties of light and matter.
		Automorphic forms is the branch of representation theory that relates symmetry with arithmetic and number theory.
	www.fields.utoronto.ca /programs/scientific/04-05/arthurconf/biography.html (559 words)

	Gelfand representation (Site not responding. Last check: 2007-10-31)
		In mathematics the Gelfand representation in functional analysis allows a complete characterisation of commutative C*-algebras as algebras of continuous complex-valued functions.
		Gelfand representation theorem is one avenue in development of spectral theory for normal operators.
		The Gelfand-Naimark theorem is a result for arbitrary (abstract) noncommutative C-algebras A which though not quite analogous to Gelfand representation* does provide a concrete representation A as an algebra of operators.
	www.freeglossary.com /Gelfand_representation (977 words)

	[No title]
		For a cuspidal representation $\pi$ of ${\rm SL}_2({\Bbb A}_E)$, we study the non-vanishing of the period integral on ${\rm SL}_2(F)\backslash {\rm SL}_2({\Bbb A}_F)$.
		Finally, we construct an automorphic representation $\pi$ on ${\rm SL}_2({\Bbb A}_E)$ which is abstractly ${\rm SL}_2({\Bbb A}_F)$ distinguished but none of the elements in the global $L$-packet determined by $\pi$ is distinguished by ${\rm SL}_2({\Bbb A}_F)$.
		It follows that if $\pi$ is a discrete series representation, then at most one of the representations $\pi$ and $\pi \otimes \omega$ is distinguished, where $\omega$ is an extension of the local class field theory character associated to $E/F$.
	www.math.tifr.res.in /~anand/publications.html (559 words)

	Representation of Lie algebras (Site not responding. Last check: 2007-10-31)
		Equivalently it's a representation of the universal enveloping algebra.
		See also group representation representations of Lie groups representation of a Lie superalgebra representation a Hopf algebra representation of an algebra representation of a group algebra representation of Lie algebra algebra representation of a Lie algebra representation of a Hopf algebra unitary of a real Lie algebra unitary representation a star Lie superalgebra.
		This book is a pretty good introduction to the theory of Lie algebras and their representations, and its importance cannot be overstated, due to the myriads of applications of Lie algebras to physics, engineering, and computer graphics.
	www.freeglossary.com /Representation_of_Lie_algebras (551 words)

	Papers by David Vogan
		As a hint, the authors represent four continents by birth and residence.) The main theorem says that Zuckerman's "A_q(lambda)" representations are isolated in the unitary dual, with a few obvious exceptions.
		The introduction to the paper (corresponding to three of the transparencies) sketches an answer to the question "what is representation theory?" that is meant to be accessible to most mathematicians.
		This is a draft of an exposition of formal aspects of formulating Kazhdan-Lusztig conjectures and Arthur's conjectures for p-adic reductive groups.
	www-math.mit.edu /~dav/paper.html (906 words)

	Robert Langlands' work - automorphics forms
		Some of their consequences were explained in a graduate course given at Princeton in the spring of 1967, and then things were put in a somewhat wider context in a series of lectures at Yale later that Spring.
		These two observations underline that functoriality arose in an attempt to find a nonabelian class-field theory under the influence of the view, which arose in the early sixties, that much of the theory of automorphic forms could and should be treated in the context of group representations.
		The formula for Whittaker functions for unramified representations suggested in the letter was proved by Casselman and Shalika.
	sunsite.ubc.ca /DigitalMathArchive/Langlands/automorphic.html (829 words)

	Interaction of Finite Dimensional Algebras with other areas of Mathematics
		Methods concerning quivers and their representations have been used in the past 30 years extensively in order to describe the structure of length categories (abelian categories where every object has a finite composition series) which arise very frequently not only in algebra, but also in geometry and analysis.
		The exciting development in the representation theory of finite dimensional algebras in the last 30 years was based on the use of very intricate combinatorial methods (quivers, root systems, posets, integral quadratic forms).
		Though it is not a principal focus of our Workshop, the connections of representation theory as specific to Lie theory and Lie groups with the general theory of representations of finite dimensional algebras and of finite groups is always present in our thinking.
	www.pims.math.ca /birs/workshops/2004/04w5501 (1176 words)

	Robert Langlands
		Robert Langlands (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with profound insights in number theory and representation theory[?].
		Langlands understood that the theory of automorphic forms gives a generalization of class field theory, a central topic in algebraic number theory.
		Thus to every representation of a Galois group there should be associated an automorphic form.
	www.fastload.org /ro/Robert_Langlands.html (374 words)

	Robert Langlands' work - main page
		He has held faculty positions at Princeton University and Yale University, and is currently a Professor at the Institute for Advanced Study in Princeton.
		He has won several awards recognizing his outstanding contributions to the theory of automorphic forms, among them an honorary degree from the University of British Columbia in 1985.
		Representation theory---its rise and its role in number theory
	sunsite.ubc.ca /DigitalMathArchive/Langlands (541 words)

	Eisenstein series bibliography
		Borel and H. Jacquet, Automorphic forms and automorphic representations, in Automorphic forms, representations and L-functions,, Proc.
		Ginzburg, A Rankin-Selberg integral for the adjoint representation of GL(3), Inv.
		R.P.Langlands, On the notion of an automorphic representation, in Proc.
	math.umn.edu /~garrett/m/v/eis_bib.html (2572 words)

	Fields Institute - Program on Automorphic Forms - Course Information
		The theory of L-functions of automorphic forms (or modular forms) via integral representations has its origin in the paper of Riemann on the zeta-function.
		L-functions of automorphic representations were first developed by Jacquet and Langlands for GL(2).
		In the context of automorphic representations, the Converse Theorem for GL(2) was developed by Jacquet and Langlands, extended and significantly strengthened to GL(3) by Jacquet, Piatetski-Shapiro, and Shalika, and then extended to GL(n).
	www.fields.utoronto.ca /programs/scientific/02-03/automorphic_forms/courses (875 words)

	André Weil (Site not responding. Last check: 2007-10-31)
		Eventually the adelic approach became basic in automorphic representation theory.
		He discovered that the so-called Weil representation, previously introduced in quantum mechanics by Irving Segal and Shale, gave a contemporary framework for understanding the classical theory of quadratic forms.
		This was also a beginning of a substantial development by others, connecting representation theory and theta-functions.
	www.tocatch.info /en/Andre_Weil.htm (1189 words)

	Fundamental representation (Site not responding. Last check: 2007-10-31)
		The irreducible representations of a simply-connected compact Lie group are indexed by their highest weight.
		Outside of Lie group theory, the term fundamental representation is sometimes loosely used to refer to a smallest-dimensional faithful representation, though this is also often called the standard or defining representation (a term referring more to the history, rather than having a well-defined mathematical meaning).
		According to the Representational Theory of Mind, psychological states are to be understood as relations between agents and mental representations.
	www.omniknow.com /common/wiki.php?in=en&term=Fundamental_representation (1718 words)

	Vignettes on automorphic and modular forms, representations, L-functions, and number theory
		Banach space representations of real reductive groups are of moderate growth.
		This implies that rho determines the isomorphism class of the (g,K)-module, and also that the whole representation space is the tensor product of the enveloping algebra U(p+) with rho.
		Standard basic features of representation theory of p-adic reductive groups: exactness of Jacquet module functors, Jacquet's lemmas, admissibility and finite-generation of Jacquet modules of admissible finitely-generated smooth representations.
	www.math.umn.edu /~garrett/m/v (1329 words)

	Automorphic form - TheBestLinks.com - Chain rule, Functional analysis, Group representation, Lie group, ...
		Automorphic form, Chain rule, Functional analysis, Group representation, Lie...
		The theory of the Selberg trace formula, as appied by others, showed the considerable depth of the theory.
		space for a quotient of the adelic form of G, an automorphic representation is a representation that is an infinite tensor product of representations of p-adic groups, with sespecific enveloping algebra representations for the infinite prime(s).
	www.thebestlinks.com /Automorphic_form.html (743 words)

	AMS Online Books/pspum31
		Representations of reductive Lie groups by N.R. Wallach
		Representations of {german p}-adic groups: A survey by P. Cartier
		Automorphic forms and automorphic....by A. Borel and H. Jacquet
	e-math.ams.org /online_bks/pspum331 (106 words)

	The Univ. of Iowa, Representation Theory and Number Theory Group
		His research interests are centered on the representation theory and harmonic analysis of p-adic groups.
		His research is focused on analytic and functorial behavior of automorphic L-functions.
		This includes bounds and zero-free regions for autmorphic L-functions, their prime number theorem, orthogonality, zero distribution, and factorization to primitive L-functions, as well as base change, automorphic induction, relative trace formulas, and the global theory in the Langlands program.
	www.math.uiowa.edu /faculty/researchGroups/grouprep.htm (107 words)

	Faculty Research Interests
		Secondly, Leonard pairs are associated with a family of orthogonal polynomials, consisting of the q-Racah polynomials and their relatives.
		The q-Racah polynomials describe the Racah coefficients for representations of the quantum algebra U_q(sl_2), and also play a role in the quantum theory of angular momentum.
		Tonghai Yang specializes in number theory, arithmetic geometry, and automorphic representation, and in particular their interaction.
	www.math.wisc.edu /~ono/algresearch.html (2328 words)

	ALGEBRAIC LIE REPRESENTATIONS
		The study of primitive spectrum of enveloping algebras and quantum groups particularly with respect to the construction of completely prime ideals, the determination of invariants such as Goldie rank and associated varieties.
		The study of geometric objects associated to the orbit method, in particular orbital varieties, their inclusion relations and character formulae, ideals of definition particularly of orbital varieties and of B-orbits.
		The study of invariant functionals on automorphic representations and of adelic variants of Frobenius reciprocity.
	www.wisdom.weizmann.ac.il /~gorelik/agrt.htm (213 words)

	[No title]
		Notes for a talk at CIRM-Luminy for the conference ``Representations du groupes p-adique Sp(4)'', June 15-18, 1998 (5 pages), pdf file (250 K).
		We show that the two main approaches to a theta correspondence for similitudes from the literature are essentially the same, and we prove that a version of strong Howe duality holds for both constructions.
		Automorphic Forms and Representations of p-adic Groups, MSRI-Banff, November 27-December 1, 2001
	www.webpages.uidaho.edu /%7Ebrooksr (895 words)

	CFT Bibliography
		I will present an unorthodox introduction to the representation theory of Lie groups and Lie algebras, focussing entirely on the group of two by two matrices with determinant one.
		Thus we hope to cover a breadth of topics in representation theory, that are usually sacrificed for the depth of treating general Lie groups.
		Wee Teck Gan, Automorphic Forms and Automorphic Representations: slides for a series of five lectures given in Hangzhou, China giving an excellent overview of the basic theory (available on his web page).
	www.math.utexas.edu /~benzvi/teaching/SL2.html (621 words)

	[No title]
		The problem is put into the context of moduli spaces of representations, in the framework of Geometric Invariant Theory.
		By tensoring the oscillator representation of the metaplectic group with the spinor representation of the spin group, a representation of the orthosymplectic Lie supergroup OSp is obtained.
		Their correlation functions are computed by considering a dual pair G x K acting inside the oscillator-spinor representation of OSp.
	www.lpt.ens.fr /BIS/20023i.html (808 words)

	Automorphic Forms and Representations
		The Theory of Automorphic Forms, rightly or wrongly, has a reputation of being difficult for the student.
		Since 1990 I have been lecturing on automorphic forms and representation theory at Stanford and the MSRI, and this book is the end result.
		Its aim is to cover a substantial portion of the theory of automorphic forms on GL(2).
	math.stanford.edu /~bump/book.html (363 words)

	Richard Pinch: Publications
		An A4~extension of Q attached to a non-selfdual automorphic form on GL(3) (with A.
		Once a motive is attached to an automorphic representation, one can derive a compatible series of $\lambda$-adic representations of $\GQ$ attached to it in the usual way.
		For a few small primes $\lambda$ in the Hecke algebra, congruences mod $\lambda$ between the computed automorphic forms and Eisenstein series coming from classical holomorphic cusp forms of weight 2 for $GL(2)$ were already noted in {\bf [AGG]}.
	www.chalcedon.demon.co.uk /rgep/publish.html (3338 words)

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