The representation theory of glpnqis relatively simple, and quite well understood now, thanks to the work of a number of mathematicians over the past forty years. To place it in perspective, we devote much of the paper to a historical introduction to the langlands program. On tensor third lfunctions of automorphic representations of glnpafq heekyoung hahn abstract. Classifying automorphic representations clay mathematics institute. Galois representations86 references 86 introduction the goal of this course is to introduce and study automorphic representations. Here is the official catalog description from cambridge. Hitchin type moduli stacks in automorphic representation theory. The purpose of the colloquium was to discuss recent achievements in the theory of automorphic forms of one and several variables, representation theory with. Representation theory and automorphic forms springerlink. An automorphic form, constructed as a nonholomorphic. The article that marc palm is mentioning has been mention to me by other people.
The area of automorphic representations is a natural continuation of studies in number theory and modular forms. Galois representations and automorphic forms mastermath. Galois representation there are huge differences between the local representation at p6. The second conference, entitled representation theory, automorphic forms, and complex geometry, was hosted by harvard university in honor of prof. Representation theory depends upon the nature of the vector space on which the algebraic object is represented. On the notion of an automorphic representation the. Classifying automorphic representations mathematical institute. Reference for automorphic form via representation theory. We shall begin with a brief description of the basic results in this theory, and then give a reformulation using representation the ory. It is good to learn modular forms or elliptic curves first, though most accounts of modular forms dont make the representation theory aspect evident. Jun 11, 2014 automorphic representations in form of induced representations with inducing quantum bundle ov er a riemann surface and then use the loop group represen tation construction to realize the. Spg reading online ajcc cancer staging manual 7th edition pdf free pdf bookboon reading online ajcc cancer staging manual 7th edi. Lfunctions and automorphic representations james arthur abstract.
Representation theory university of california, berkeley. Then r arises from a cuspidal automorphic representation. These are notes from courses on automorphic representations given by jayce r. The advantage of elliptic curves is that you will probably see the representation theory sooner. Iv automorphic representations 27 basic representation theory of real and padic groups. Shahidis paper intertwining operators, lfunctions, and representation theory, notes of the eleventh kaist mathematics workshop 1996, taejon, korea. Automorphic representations and galois representations. Workshop automorphic galois representations, lfunctions and. Representation theory relates to number theory through the langlands program, which conjecturally connects algebraic extensions of number fields to automorphic representations and lfunctions. The invited papers, written by leading mathematicians, track recent progress in the ever expanding fields of representation theory and automorphic forms, and their association with. These conjectures led to the theory of modular forms. I will present hida theory for automorphic forms and its application to the study of the eisenstein ideal.
In the language of representation theory one of the fundamental problems of the modern theory of automorphic forms is the following. This book, published by cambridge university press, is now shipping. We have a satisfactory notion of automorphic form when r c. Definition of automorphic representation mathematics. Wallach 1 orbital integrals and the harishchandra transform. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations with finite dimensional galois representations.
This volume addresses the interplay between representation theory and automorphic forms. The subsequent notion of an automorphic representation has proved of great technical value when dealing with g an algebraic group, treated as an adelic algebraic group. The above theory of hecke works beautifully for modular forms with respect to sl2. Conjecture langlands for any continuous representation of the galois group there is an automorphic representation tt 6 iig, neces sarily unique, such that ova cr,r for all v outside some finite set s 2 sm. Automorphic forms, representations, and combinatorics.
Workshop automorphic galois representations, lfunctions. It is not used in the hecke theory but in the chapter on automorphic forms and quaternion algebras. We give two examples that will later turn out to be known special cases of the langlands correspondence, namely gausss quadratic reciprocity theorem and the modularity theorem of wiles et al. The book features extensive foundational material on the representation theory of gl 1 and gl 2 over local fields, the theory of automorphic representations, lfunctions and advanced topics such as the langlands conjectures, the weil representation, the rankinselberg method and the triple lfunction, and examines this subject matter from many. G be the center, so that z is isomorphic to the multiplicative group. The third chapter is devoted to the local theory of galois representations. Representation theory, automorphic forms, and complex geometry a conference in honor of the 70th birthday of wilfried schmid harvard university, science center hall c, may 2023, 20. G can be embedded in gl n in such a way that it is stable under transpose. Kowalski, section 3 of automorphic forms, lfunctions and number theory march 1216 three introductory lectures dorian goldfeld, joseph hundley, chapter 2 of automorphic representations and lfunctions for the general linear group, cambridge studies in advanced mathematics 129, 2011 pdf. Database of automorphic lfunctions rutgers university.
Automorphic forms and automorphic representations 189 200. Zhang shtukas and the taylor expansion of lfunctions ii. For representation theory of finite groups, the classic is serres linear representations of finite groups. It does not completely include the automorphic form idea introduced above, in that the adelic approach is a way of dealing with the whole family of congruence subgroups at once. The book contains a wealth of information about discrete subgroups and automorphic representations, and can be used both as a very good introduction to the subject and as a valuable reference. Eisenstein series and automorphic representations the moduli space of super riemann surfaces since one is dealing with superstring theory and it is known that this integral cannot be reduced in a simple manner to an integral over ordinary riemann surfaces for h5 111. There are several different classes of groups, associative algebras and lie algebras, and their representation theories all have an individual flavour.
Theres a lot involved in automorphic forms, and a lot of aspects to come at it from. We study these problems in a setting related to the langlands lfunctions lps. However, no body has ever been able to point out the precise definition or theorem or anything concrete. Automorphic forms and automorphic representations wee teck gan references. The latter control the embedding of a model of a unitary representation into spaces ofautomorphicfunctions,andwere earlier studiedby ehrenpreis. Automorphic forms, representation theory and arithmetic. The basic example is g acting on l2g by left or right translation. In the mid1990s wilfried schmid became interested in applying analytic concepts from representation theory to automorphic forms via their boundary value distributions. In this theory, one considers representations of the group algebra a cg of a.
Jan 07, 2020 the second conference, entitled representation theory, automorphic forms, and complex geometry, was hosted by harvard university in honor of prof. Decomposition of representations into tensor products 179 190. Aninternationalcolloquiumon automorphic forms, representation theory and arithmetic was held at the tata institute of fundamental research, bombay, from 8 to 15 january 1979. We note that the general langlands correspondence is still largely. Representation theory and automorphic forms toshiyuki. We provide an introduction to the theory of eisenstein series and automorphic forms on real simple lie groups g, emphasising the role of representation theory. The conjecture represents a fundamental problem in number theory. Given a global eld fand a reductive algebraic group gover f, then an automorphic representation of g is a g. An introduction to automorphic representations duke mathematics. We shall begin with a brief description of the basic results in this theory, and then give a reformulation using representation theory.
Representation theory depends upon the type of algebraic object being represented. Daniel bump has been a leading mathematician in automorphic forms, representation theory. Classify all representations of a given group g, up to isomorphism. The theory of automorphic forms then shows that ls. If g is a topological group, then a unitary representation of g is an isometric action of g on a hilbert space h so that the action map g. Representation theory of the latter group naturally leads to the study of automorphic functions and related numbertheoretic problems.
Representation theory and automorphic forms american. Our goal is to formulate a theorem that is part of a recent classi cation of automorphic representations of orthogonal and symplectic groups. The conference automorphic forms, representations, and combinatorics will take place at stanford university from august 16, 2012. Pdf fourier coefficients and small automorphic representations. Using these automorphic forms, one can construct the automorphic representation associated to f. To place it in perspective, we devote much of the paper.
The book features extensive foundational material on the representation theory of gl1 and gl2 over local fields, the theory of automorphic representations, lfunctions and advanced topics such as the langlands conjectures, the weil representation, the rankinselberg method and the triple lfunction, examining this subject matter from many. The invited papers, written by leading mathematicians, track recent progress in the ever expanding fields of representation theory and automorphic forms, and their association with number theory and. Knapp, editors, representation theory and automorphic forms international centre for mathematical sciences. Automorphy for some ladic lifts of automorphic mod l galois. Automorphic forms and representations by daniel bump. Introductory lectures on automorphic forms lectures for the european school of group theory july, 2001, luminy, france by nolan r.
Representation theory, automorphic forms, and complex geometry. Given an automorphic form f, roughly speaking, one considers the vector space v. Langlands beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. This reformulation leads to a vast generalization of the theory.
386 1217 382 861 653 1120 466 1288 1413 123 994 1223 595 249 388 1507 14 1509 387 934 701 436 160 287 1430 218 701 443 1288 1436 1075 809 648 576 8 1027 396