langlands correspondence for loop groups

There are numerous variations of this, in part because the definitions of Langlands group and L … Log In with Facebook Log In with Google. 1.4.6 Loop groups Derived Categories in Geometrical Langlands Ramifications Problem. Study Group on a Langlands Correspondence for Loop Groups. It provides an example of the geometric Langlands correspondence with wild ramification. Abstract: It is expected that, under mild conditions, local Langlands correspondence preserves depths of representations. (i) One begins with the Langlands dual group and a semistable homomorphism . Vertex Algebras and Algebraic Curves (joint with David Ben-Zvi ) Second Edition was published by the … A family of representations of a ne Lie algebras (with B. Feigin), Russ. Thus, the Langlands correspondence for loop groups is categorical: we associate categorical representations of G..t//to local Langlands parameters. Langlands Correspondence for Loop Groups has been published by Cambridge University Press in June 2007. Read Langlands Correspondence for Loop Groups: 103 (Cambridge Studies in Advanced Mathematics) book reviews & author details and more at Amazon.in. This construction aims to attach to each Langlands … a combinatorial model for U_q(Ln), and connect it with Enriquez' degeneration A of the elliptic algebras of Feigin-Odesskii. In the traditional local Langlands correspondence one wishes to describe smooth representations of a reductive group over a local non-archimedian field, such as the field of formal Laurent power series over a finite field, in terms of the Galois group of the local field and the Langlands dual group. Introduction The theory of quantum groups emerged in mid 1980s from the work of St. Petersburg math- In 2012, Vincent Lafforgue established the global Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields. Philip Kutzko ( 1980) proved the local Langlands conjectures for the general linear group GL (2, K) over local fields. In mathematics, the local Langlands conjectures, introduced by Robert Langlands ( 1967, 1970 ), 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. Geometric Satake correspondence (will be introduced, necessary for the talks 3.2 and 5.1-5.3) Automorphic functions on adele groups (will be introduced; serves as crucial piece of motivation) Loop groups (will be introduced, necessary for several talks) Factorization algebras (will be introduced; serve as motivation for talks on factorization) Would one need representation theory, elements of algebraic number theory ,modular forms … Classical limit of the geometric Langlands correspondence. Geometric Satake correspondence (will be introduced, necessary for the talks 3.2 and 5.1-5.3) Automorphic functions on adele groups (will be introduced; serves as crucial piece of motivation) Loop groups (will be introduced, necessary for several talks) Factorization algebras (will be introduced; serve as motivation for talks on factorization) Local geometric Langlands correspondence and affine Kac-Moody algebras . Aswin Balasubramanian Geometric Langlands from 4d N … Geometric Langlands correspondence in the restricted setting. Epipelagic representations and rigid local systems 1.4.5 Langlands dual group Let G denote the reductive group over Q whose root system is dual to that of G. We also ﬁx a pinning † of G †, through which we identify Aut (G )with Aut†(G).We deﬁne the Langlands dual group LG of G to be LG = G μe where μe acts through μe → Aut†(G)∼= Aut†(G). The Langlands correspondence for higher local fields is still at an early stage of development. Corresponds to Sect. An important new element in our proposal is that, in contrast to the classical correspondence, the loop group now acts on categories rather than vector spaces. its analogs (real forms, loop group etc) would appear and functions on a Riemann surface C replace the global eld Z. Corresponds to Sect. March 24 . What is called the Langlands correspondence in number theory ( Langlands 67) is first of all a conjectural correspondence (a bijection subject to various conditions) between. The local Langlands conjectures are certain conjectures in the context of the Langlands program. A candidate for the local geometric Langlands correspondence is constructed by Frenkel and Gaitsgory in [11]. The local geometric Langlands parameters associated to the loop group of g are given by the space Conn g (D ) of g local systems on D . MR 4. Vertex Algebras and Algebraic Curves (joint with David Ben-Zvi ) Second Edition was published by the … Appl. Free delivery on qualified orders. The theory offers fresh insights into the world of Langlands dualities, with many applications to Representation Theory of Infinite-dimensional Algebras, and Quantum Field Theory. See more Langlands Correspondence for Loop Groups by Ed... Email to friends Share on Facebook - opens in a new window or tab Share on Twitter - opens in a new window or tab Asking Langlands Correspondence For Loop Groups (Cambridge Studies In Advanced Mathematics) Edward Frenkel for help with an essay to professionals from the portal , you are guaranteed to get the help that is necessary for you and Langlands Correspondence For Loop Groups (Cambridge Studies In Advanced Mathematics) Edward Frenkel your scientific material. Math. The Drinfeld-Langlands correspondence, also called the geometric Langlands correspon-dence, is a conjecture analogous to the Langlands correspondence for a reductive group over a nite extension Fof k(t), where kis an arbitrary eld. The local Langlands correspondence connects representation of p-adic groups to Langlands parameters, which are certain representations of Galois groups of local fields. Email: Password: Remember me on this computer. E.g., Frenkel's book "Langlands Correspondence for Loop Groups". Reciprocity laws Let p be a prime number 1 modulo 4. B. Sury The Langlands Program. Introduction. ... _p$. No seminar on Monday. The conjecture is unresolved even for GL2 which we shall now elaborate. Representations of the absolute Galois group correspond to modular objects on the moduli space of curves. To summarize, our strategy [FG2] for constructing the global geometric Langlands correspondence has two parts: Ramiﬁcations of the Geometric Langlands Program 55 (1)the local part: describing the structure of the categories of g-modules of The theory offers fresh insights into the world of Langlands dualities, with many applications to Representation Theory of Infinite-dimensional Algebras, and Quantum Field Theory. The loop realization of the Yangian 40 5.10. Description: This book provides an excellent detailed review of an important aspect of the geometric Langlands program, namely, the role of representation theory of affine Kac-Moody algebras (or loop algebras). Chiral algebra tools methods for the analytic Geometric Langlands program, continuation ... character sheaves on the loop group, conclusion ; Link to the recording ; September 16 . With this series of books, the authors have provided a selection of worked examples, problems with complete solutions and test papers designed to be used with or instead of standard textbooks on algebra. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. MR 2332156 [17] The loop polarization 36 5.7. 2. We would like to show you a description here but the site won’t allow us. The local geometric Langlands philosophy for loop groups (after Frenkel{Gaitsgory) By the Langlands philosophy, the space of Langlands parameters should be a space of local systems, i.e. generalizing local class field theory from abelian Galois groups to non-abelian Galois groups. Langlands Correspondence for Loop Groups has been published by Cambridge University Press in June 2007. In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. We prove half of this conjecture by showing that the depth of a categorical representation of the loop group is less than or equal to the depth of its underlying geometric Langlands parameter. Sign Up with Apple. $\endgroup$ – Marty Jun 26 '18 at 3:40 1 March 24 . The Langlands dual group certainly arises in these studies, but I haven't seen something quite as straightforward as a parameters from the Weil group (described above) to the dual group. In addition to his New York Times bestseller Love and Math, Frenkel has published two other books: Langlands Correspondence for Loop Groups and Vertex Algebras and Algebraic Curves (with David Ben-Zvi). Langlands Correspondence for Loop Groups The Langlands Program was conceived initially as a bridge between Number Theory and Automorphic Representations, and has now expanded into such areas as Geometry and Quantum Field Theory, weaving together seemingly unrelated disciplines into a web of tantalizing conjectures. A candidate for the local geometric Langlands correspondence is constructed by Frenkel and Gaitsgory in [11]. Abstract The classical limit of the geometric Langlands correspondence is the conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a reductive group G and its dual. Papers 1. 4) What is the Local Langlands correspondence, explicitly, for small values of n? 5) What is the Jacquet-Langlands theorem, vaguely speaking? To summarize, our strategy [FG2] for constructing the global geometric Langlands correspondence has two parts: Ramiﬁcations of the Geometric Langlands Program 55 (1)the local part: describing the structure of the categories of g-modules of After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry group LG. × Close Log In. An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. This construction aims to attach to each Langlands … Cohomology of the commutator subgroup of the braid group, Funct. there should be a universal category C, bred over LocSys g_(D ), such that the following diagram commutes: bg-mod crit / Op bg_(D ) C /LocSys g_(D ) Idea 0.1. Hello Select your address Best Sellers Today's Deals New Releases Electronics Books Customer Service Gift Ideas Home Computers Gift Cards Sell These $\ell$-adic representations, and their characteristic $0$ counterparts, have been constructed in some cases by Deligne and Katz. Read Langlands Correspondence for Loop Groups: 103 (Cambridge Studies in Advanced Mathematics) book reviews & author details and more at Amazon.in. geometric Langlands correspondence. Langlands program: V. Lafforgue proved Langlands correspondence for reductive groups over function ﬁelds. Number of pages: 393. Free 2-day shipping. Tuesday 30 October 2007, 16:00-18:00 Enter the email address you signed up with and we'll email you a reset link. Buy Cambridge Studies in Advanced Mathematics (Hardcover): Langlands Correspondence for Loop Groups (Series #103) (Hardcover) at Walmart.com دانلود کتاب Langlands correspondence for loop groups به فارسی مکاتبات Langlands برای گروه های حلقه ای حجم 2 MB فرمت pdf تعداد صفحات 396 سال نشر 2007 :نویسنده Edward Frenkel :ناشر CUP It turns out that there is indeed an analogue of the local Langlands correspondence for the loop groups, but that one should consider their representations on categories, rather than vector spaces. The Geometric Langlands Correspondence can be formulated as follows. The ultimate form of the local Langlands correspondence for loop groups should be, roughly, the following statement: (1.1) categories ﬁbering over LocGˇ(D×) ⇐⇒ categories equipped with action of G((t)) We should point out, however, that neither the notion ofcategoryﬁbered overa non-algebraic It provides an example of the geometric Langlands correspondence with wild ramification. of Wisconsin). Local Langlands correspondence (Vogan, Bernstein, Kaletha): Better to study representation theory of G(F) together with that of its extended pure inner forms. n - dimensional complex linear representations of the Galois group Gal(ˉF / F) of a given number field F, and. Beginning with the more familiar setting of Picard groups, we will explore the principal structures on these spaces, the loop group uniformization and Hitchin's integrable system. Such representations can be realized in spaces of global sections of twisted D-modules on the quotient of the loop group G((t)) by its “open compact” subgroup K, such as G[[t]] or the Iwahori subgroup I. In mathematics, the local Langlands conjectures, introduced by Robert Langlands (1967, 1970), 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. Description xvi, 379 p. : ill. ; 24 cm. Series Cambridge studies in advanced mathematics ; v. 103 Notes Includes bibliographical references and index. number theory: Xiao-Zhu proved the ”generic” case of the Tate conjecture on the mod p ﬁbres of some Shimura varieties. 5.6. ... (2007) Langlands Correspondence for Loop Groups. E. Frenkel: Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics 103, Cambridge University Press 2007, ISBN 978-0-521-85443-6. Day 5: Lecture 5.1: Langlands duality for local and global Whittaker categories. 3. Anal. Over R, I recall seeing something by Nadler or Ben-Zvi. Hence the appearance of words like "loop group" and "Kac-Moody group". In this article, we formulate a conjectural geometrisation of this expectation. Fully hyperlinked electronic version of the book is available here. Davide Gaiotto. An important new element in our proposal is that, in contrast to the classical correspondence, the loop group now acts on categories rather than vector spaces. Thus, the Langlands correspondence for loop groups is categorical: we associate categorical representations of G..t//to local Langlands parameters. Dima Arinkin (Univ. Langlands Correspondence for Loop Groups by Edward Frenkel, 9780521854436, available at Book Depository with free delivery worldwide. Geometric Langlands correspondence in the restricted setting. ISBN/ASIN: 0521854431. Math. Lyndon words and quantum loop groups. ISBN-13: 9780521854436. Where the genuine Langlands correspondence concerns global fields, the local Langlands correspondence concerns local fields. or reset password. This course will discuss one of the most exciting recent development in automorphic forms and number theory: the paper (still under revision) Chtoucas pour les groupes reductifs et parametrisation de Langlands globale of Vincent Lafforgue on the global Langlands correspondence over function fields. Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L-group. Categories acted on by the loop group We will assume having the following notions at our disposal: 1.1. The theory offers fresh insights into the world of Langlands dualities, with many applications to Representation Theory of Infinite-dimensional Algebras, and Quantum Field Theory. Langlands Correspondence for Loop Groups ICM Edition Edward Frenkel — 2010-07-23 in Langlands Correspondence For Loop Groups (Cambridge Studies In Advanced Mathematics) Edward Frenkel And if we cooperate with professionals, then only with the authors from the Langlands Correspondence For Loop Groups (Cambridge Studies In Advanced Mathematics) Edward Frenkel portal . Amazon.in - Buy Langlands Correspondence for Loop Groups: 103 (Cambridge Studies in Advanced Mathematics) book online at best prices in India on Amazon.in. Day 5: Lecture 5.1: Langlands duality for local and global Whittaker categories. Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory Meng-Chwan Tan Department of Physics, National University of … The Langlands correspondence for classical groups has recently been established by Arthur. If Xis a quasi-projective smooth curve over kwith function eld F, then the correspondence states a conjectural Edward Frenkel, Langlands correspondence for loop groups, Cambridge Studies in Advanced Mathematics, vol. On refined Chern-Simons and refined ABJ matrix models [12mm] Luca Cassia luca.cassia@physics.uu.se and Maxim Zabzine maxim.zabzine@physics.uu.se [8mm] Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden. (still under revision) Chtoucas pour les groupes reductifs et parametrisation de Langlands globale of Vincent Lafforgue Abstract. Langlands correspondence for loop groups / Edward Frenkel. Surv. The ultimate form of the local Langlands correspondence for loop groups should be, roughly, th~ following statement: (1.1) categories fibering over Loc0 ( 1) x ) categories equipped with action of G((t)) We should point out, however, that neither the notion of category fibered over a non-algebraic stack such as Loc0 (1)x), nor the unversal or. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. half of a quantum group. Subject headings Loops (Group theory) ISBN These $\ell$-adic representations, and their characteristic $0$ counterparts, have been constructed in some cases by Deligne and Katz. 3) What is the Local Langlands correspondence? 1 Remember that this is a correspondence between flat holomorphic bundles of the worldsheet (hypersurface) and Hecke eigensheaves on the moduli space of holomorphic bundles on where. It is expected that, under mild conditions, local Langlands correspondence preserves depths of representations. He has also authored over 90 scholarly articles in academic journals, and he has lectured on his work around the world. It is advertised as an undergraduate course, but it seems this would apply to either very dedicated or quite advanced students after reading the first pages. Contents: a) Definition of the local Whittaker category We define a loop version of this construction, i.e. Chiral algebra tools methods for the analytic Geometric Langlands program, continuation ... character sheaves on the loop group, conclusion ; Link to the recording ; September 16 . Introduction. Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathe-matics 103, Cambridge University Press, 2007 (395 pp.). S. Arkhipov, R. Bezrukavnikov and V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, Journal of AMS 17 (2004) 595-678. zbMATH MathSciNet Google Scholar [AG] S. Arkhipov and D. Gaitsgory, Another realization of the category of modules over the small quantum group, Adv. 10.1 of the Outline. In the Langlands Program, one development is a very general trace formula due to James Arthur and his collaborators. 103, Cambridge University Press, Cambridge, 2007. In recent years [1] [2], it was realized that the Langlands correspondence is the number-theoretical counterpart of the electromagnetic duality in gauge field theories. Prerequisites for E. Frenkel's “Langlands correspondance for loop groups”. E. Frenkel and D. Ben-Zvi: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, Second Edition, American Mathematical Society 2004, ISBN 0-8218-3674-9. Idea 0.1. The roots of the polynomials f = X2 p and g = Xp 1 generate elds K f = Q(p p) and K g = Q( p) where Edward Frenkel has put a draft version of his new book Langlands Correspondence for Loop Groups on his web-site. of G((t)). It is partially based on the graduate courses taught by … Langlands Correspondence For Loop Groups really liked it 4.00 avg rating — 2 ratings — published 2007 — 3 editions Want to Read saving… From its early days, some of the program’s statements were also known to also haverepresentation theoreticconsequences. The notion of category C acted on by the loop group G((t)) at level c(for an abelian category this notion is developed, e.g., in [FG2]). 2. Abstract: An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange The ultimate form of the local Langlands correspondence for loop groups should be, roughly, th~ following statement: (1.1) categories fibering over Loc0 ( 1) x ) categories equipped with action of G((t)) We should point out, however, that neither the notion of category fibered over a non-algebraic stack such as Loc0 (1)x), nor the unversal Atiyah was born on 22 April 1929 in Hampstead, London, England, the son of Jean (née Levens) and Edward Atiyah. 10.1 of the Outline. Hence the appearance of words like "loop group" and "Kac-Moody group". Thursday (March 10), 4:30 p.m, room E 206. In the seminal paper [ ] , Witten gave a gauge field theory construction of the celebrated Jones polynomials … Publisher: Cambridge University Press 2007. Notation. This will be the main theme of the present paper. 22, 248{250 (1988). The goal of this book is to present a systematic and self-contained introduction to the local geometric Langlands Correspondence for loop groups and the related representation theory of affine Kac-Moody algebras. If Xis a quasi-projective smooth curve over kwith function eld F, then the correspondence states a conjectural symmetry group LG. UUITP-32/21. Amazon.in - Buy Langlands Correspondence for Loop Groups: 103 (Cambridge Studies in Advanced Mathematics) book online at best prices in India on Amazon.in. It develops the geometric Langlands Correspondence for Loop Groups, a new approach, from a unique perspective offered by affine Kac-Moody algebras. It develops the geometric Langlands Correspondence for Loop Groups, a new approach, from a unique perspective offered by affine Kac-Moody algebras. Davide Gaiotto. Contents: a) Definition of the local Whittaker category (U. of Chicago Geometric Langlands … His mother was Scottish and his father was a Lebanese Orthodox Christian.He had two brothers, Patrick (deceased) and Joe, and a sister, … these representations of loop groups. Shuffle algebras provide combinatorial models for U_q(n), i.e. The local geometric Langlands parameters associated to the loop group of g are given by the space Conn g (D ) of g local systems on D . Langlands correspondence for loop groups. The construction is applied to the study of the topology of the affine Grassmannian of G and to establishing a Langlands type correspondence for "automorphic" sheaves … This duality may also be stated purely in terms of semi-simple Lie algebras. The Drinfeld-Langlands correspondence, also called the geometric Langlands correspon-dence, is a conjecture analogous to the Langlands correspondence for a reductive group over a nite extension Fof k(t), where kis an arbitrary eld. Free delivery on qualified orders. Password requirements: 6 to 30 characters long; ASCII characters only (characters found on a standard US keyboard); must contain at least 4 different symbols; 01/26/2016. The Langlands Program was conceived initially as a bridge between Number Theory and Automorphic Representations, and has now expanded into such areas as Geometry and Quantum Field Theory, tying together seemingly unrelated disciplines into a web of tantalizing conjectures. The Yangian of an arbitrary simple Lie algebra 40 5.11. q-characters 41 References 42 1. Fully hyperlinked electronic version of the book is available here. The setting for the geometric Langlands program is the geometry of moduli spaces of bundles on an algebraic curve. Geometric Langlands (Drinfeld, Gaitsgory): the "local Langlands category" should be the categorical Frobenius xed points of the category of Q ‘-sheaves on the loop group of G. On the Langlands Correspondence in the de Rham Setting, I and II (MSRI Geometric Langlands conference, 3/02) Topological Epsilon Factors (Luminy Langlands Program meeting, 6/06) ... Lecture Series on Beilinson-Bernstein Localization for Loop Groups 39pp. The Faddeev-Reshetikhin-Takhtajan formalism 36 5.8. We refer to [56] for a discussion f Infinite Dimensional Groups and Automorphic L–Functions 685 of the conjecture for more general groups and to [3, 4, 8, 9, 30] for the original papers. Langlands Correspondence for Loop Groupsby Edward Frenkel. It develops the geometric Langlands Correspondence for Loop Groups, a new approach, from a unique perspective offered by affine Kac-Moody algebras. Prerequisites: Loop groups, BB localization (assumed), D-modules on algebraic stacks, affine BB localization (will be sketched). In this article, we formulate a conjectural geometrisation of this expectation. Format Book Published Cambridge : Cambridge University Press, 2007. I haven't really kept up with it, but here's some key points. The representations of GL n ( F) appearing in the local Langlands correspondence are smooth irreducible complex representations. "Smooth" means that every vector is fixed by some open subgroup. "Irreducible" means that the representation is nonzero and has no subrepresentations other than 0 and itself. The first account of local geometric Langlands Correspondence, a new area of mathematical physics developed by the author. In the traditional local Langlands correspondence one wishes to describe smooth representations of a reductive group over a local non-archimedian field, such as the field of formal Laurent power series over a finite field, in terms of the Galois group of the local field and the Langlands dual group. Deligne constructed a remarkable local system on $\mathbb{P}^1-\{0,\infty\}$ attached to a family of Kloosterman sums. x0. As Arthur's arguments are global in nature, one might want to know explicitly what the local correspondence is. What would be the prerequisites to understand the material in this book ? We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. Problem-solving is an art central to understanding and ability in mathematics. The centre of the vertex algebra attached to the central extension of sl_2((t)) Speaker to be confirmed. Langlands correspondence for loop groups. The (conjectural) Langlands correspondence for number fields gave rise to a Langlands correspondence for function fields (proved by Drinfeld and Lafforgue), where the most important difference is the absence of the infinite place which simplifies things in the latter. The main ideas of the Geometric Langlands Correspondence is now summerized and compaired with our construction in order to show that the same automorphic representations are obtained by the both methods. Prerequisites: Loop groups, BB localization (assumed), D-modules on algebraic stacks, affine BB localization (will be sketched). We prove half of this conjecture by showing that the depth of a categorical representation of the loop group is less than or equal to the depth of its underlying geometric Langlands parameter. 1. (5) ‘t Hooft operators of gauge theory correspond to the Hecke operators of the ... representation R of the gauge group G. In particular, a Wilson loop is labeled by a representation R of G. By contrast, the ‘t Hooft operator is a “disorder operator.” It is defined not by Yangians 39 5.9.

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