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R. P. Langlands refers to Robert Phelan Langlands, a Canadian mathematician born in 1936. He is best known for launching the Langlands Program, an influential set of conjectures and results that connect number theory with representation theory, algebraic geometry, and related fields. Langlands has received numerous honors for his work, including the Wolf Prize and the Abel Prize, reflecting the profound impact of his ideas on modern mathematics.

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Automorphic Forms on GL (2)

1970-01-01
H. Jacquet , R. P. Langlands
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On the Functional Equations Satisfied by Eisenstein Series

1976-01-01
R. P. Langlands
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On the classification of irreducible representations of real algebraic groups

1989-01-01
R. P. Langlands
Suppose G is a connected reductive group over a global field F . Many of the problems of the theory of automorphic forms involve some aspect of study of the … Suppose G is a connected reductive group over a global field F . Many of the problems of the theory of automorphic forms involve some aspect of study of the representation ρ of G(A(F )) on the space of slowly increasing functions on the homogeneous space G(F )\G(A(F )). It is of particular interest to study the irreducible constituents of ρ. In a lecture [9], published some time ago, but unfortunately rendered difficult to read by a number of small errors and a general imprecision, reflections in part of a hastiness for which my excitement at the time may be to blame, I formulated some questions about these constituents which seemed to me then, as they do today, of some fascination. The questions have analogues when F is a local field; these concern the irreducible admissible representations of G(F ). As I remarked in the lecture, there are cases in which the answers to the questions are implicit in existing theories. If G is abelian they are consequences of class field theory, especially of the Tate-Nakayama duality. This is verified in [10]. If F is the real or complex field, they are consequences of the results obtained by Harish-Chandra for representations ∗Preprint, Institute for Advanced Study, 1973. Appeared in Math. Surveys and Monographs, No. 31, AMS (1988)

BASE CHANGE FOR GL(2)

1980-01-01
R. P. Langlands
R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a … R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic representation of the group GL(2, /A), /A being the adele ring of the field, and L(s, ?), whose definition is ultimately due to Hecke, is known to be entire. The main result, from which the existence of ? follows, is that it is always possible to transfer automorphic representations of GL(2) over one number field to representations over a cyclic extension of the field. The tools he employs here are the trace formula and harmonic analysis on the group GL(2) over a local field.

<i>L</i>-Indistinguishability For <i>SL</i> (2)

1979-08-01
J.-P. Labesse , R. P. Langlands
The notion of L -indistinguishability, like many others current in the study of L -functions, has yet to be completely defined, but it is in our opinion important for the … The notion of L -indistinguishability, like many others current in the study of L -functions, has yet to be completely defined, but it is in our opinion important for the study of automorphic forms and of representations of algebraic groups. In this paper we study it for the simplest class of groups, basically forms of SL (2). Although the definition we use is applicable to very few groups, there is every reason to believe that the results will have general analogues [ 12 ]. The phenomena which the notion is intended to express have been met–and exploited–by others (Hecke [ 5 ] § 13, Shimura [ 17 ]). Their source seems to lie in the distinction between conjugacy and stable conjugacy. If F is a field, G a reductive algebraic group over F, and the algebraic closure of F then two elements of G(F) may be conjugate in without being conjugate in G(F) .
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Conformal invariance in two-dimensional percolation

1994-01-01
R. P. Langlands , Philippe Pouliot , Yvan Saint-Aubin
The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it … The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the seventeenth century. It was introduced more recently into mathematics by S. R. Broadbent and J. M. Hammersley ([BH]) and is a branch of probability theory that is especially close to statistical mechanics. Broadbent and Hammersley distinguish between two types of spreading of a fluid through a medium, or between two aspects of the probabilistic models of such processes: diffusion processes, in which the random mechanism is ascribed to the fluid; and percolation processes, in which it is ascribed to the medium. A percolation process typically depends on one or more probabilistic parameters. For example, if molecules of a gas are absorbed at the surface of a porous solid (as in a gas mask) then their ability to penetrate the solid depends on the sizes of the pores in it and their positions, both conceived to be distributed in some random manner. A simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner (that could be determined by a periodic graph), and to be open (thus very large) or closed (thus smaller than the molecules) with probabilities p and 1 − p. As p increases the probability of deeper penetration of the gas into the interior of the solid grows.
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On the definition of transfer factors

1987-03-01
R. P. Langlands , D. Shelstad

On the universality of crossing probabilities in two-dimensional percolation

1992-05-01
R. P. Langlands , C. Pichet , Ph. Pouliot +1 more
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Problems in the theory of automorphic forms to Salomon Bochner in gratitude

1970-01-01
R. P. Langlands

Automorphic representations, Shimura varieties, and motives. Ein Märchen

1979-01-01
R. P. Langlands
1. Introduction. It had been my intention to survey the problems posed by the study of zetafunctions of Shimura varieties. But I was too sanguine. This would be a mammoth … 1. Introduction. It had been my intention to survey the problems posed by the study of zetafunctions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representationtheoretic, but they both arise in the study of the zeta-function at the infinite places. The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too imprecise, and were not even in principle amenable to proof by Shimura’s methods of descent. The conjecture as it stands is the only statement I could discover that meets his criticism and is compatible with Shimura’s conjecture. The statement of the conjecture must be preceded by some constructions, which have implications that had escaped me. When combined with Deligne’s conception of Shimura varieties as parameter varieties for families of motives they suggest the introduction of a group, here called the Taniyama group, which may be of importance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed. With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of formulating one or two specific conjectures, but we begin to weave a tissue of surmise and hypothesis, and curiosity drives us on. Deligne’s ideas are reviewed in the fourth section, but to understand them one must be familiar at least with the elements of the formalism of tannakian categories underlying the conjectural theory of motives, say, with the main results of Chapter II of [40]. The present Summer Institute is predicated on the belief that there is a close relation between automorphic representations and motives. The relation is usually couched in terms of L-functions, * First appeared in Automorphic forms, representations, and L-functions, Proc. of Symp. in Pure

The Dimension of Spaces of Automorphic Forms

1963-01-01
R. P. Langlands
1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the … 1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in iarish-Chandra's investigations of the representations of semi-simple groups. These integrals have been evaluated. In this paper, after clarifying the relation between the two types of integrals, we go on to complete the evaluation of the integrals appearing in the trace formula. Before the formula for the dimension that results is described let us review iarish-Chandra's construction of bounded symmetric domains and introduce the automorphic forms to be considered. If (7 is the connected component of the identity in the group of pseudoconformal mappings of a bounded symmetric domain then (7 has a trivial centre and a maximal compact subgroup of any simple component has nondiscrete centre. Conversely if (7 is a connected semi-simple group with these two properties then (7 is the connected component of the identity in the group of pseudo-conformal mappings of a bounded symmetric domain [2(d)]. Let g be the Lie algebra of (7 and gc its complexification. Let GC be the simplyconnected complex Lie group with Lie algebra gc; replace G by the connected subgroup G of GC with Lie algebra g. Let K be a maximal compact sub
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A Combinatorial Laplacian with Vertex Weights

1996-08-01
Fan Chung , R. P. Langlands
One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of … One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see [1, 17, 11, 15]). The usual notion of the combinatorial Laplacian for a graph involves edge weights. Namely, a Laplacian L forGis a matrix with rows and columns indexed by the vertex setVofG, and the (u,v)-entry of L, foru,vinG,u≠v, is associated with the edge-weight of the edge (u,v). It is not so obvious to consider Laplacians with vertex weights (except for using some symmetric combinations of the vertex weights to define edge-weights). In this note, we consider a vertex weighted Laplacian which is motivated by a problem arising in the study of algebro-geometric aspects of the Bethe Ansatz [18]. The usual Laplacian can be regarded as a special case with all vertex-weights equal. We will generalize the matrix-tree theorem to matrix-tree theorems of counting "rooted" directed spanning trees. In addition, the characteristic polynomial of the vertex-weighted Laplacian has coefficients with similar interpretations. We also consider subgraphs with non-trial boundary. We will shown that the Laplacian with Dirichlet boundary condition has its determinant equal to the number of rooted spanning forests. The usual matrix-tree theorem is a special case with the boundary consisting of one single vertex.

Lectures in Modern Analysis and Applications III

1970-01-01
R. M. Dudley , J. Feldman , B. Kostant +2 more
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Modular Forms and ℓ-Adic Representations

2009-02-27
R. P. Langlands

Stable Conjugacy: Definitions and Lemmas

1979-08-01
R. P. Langlands
The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and … The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious. The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.
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Representations of abelian algebraic groups

1997-12-01
R. P. Langlands
There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive … There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension
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Descent for Transfer Factors

2007-01-01
R. P. Langlands , D. Shelstad
In [] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do … In [] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do so in the present paper. Nonetheless we carry out what is probably an unavoidable step in any proof of existence: reduction to a local statement at the identity in the centralizer of a semisimple element, a favorite procedure of Harish Chandra that he referred to as descent.

Dimension of spaces of automorphic forms

1966-01-01
R. P. Langlands
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On the Zeta-Functions of Some Simple Shimura Varieties

1979-12-01
R. P. Langlands
In an earlier paper [ 14 ] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real … In an earlier paper [ 14 ] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [ 13 ]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [ 13 ] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [ 17 ].
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Algebro-geometric aspects of the Bethe equations

2008-04-06
R. P. Langlands , Yvan Saint-Aubin

The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups

1966-01-01
R. P. Langlands

Orbital Integrals on Forms of SL(3), I

1983-04-01
R. P. Langlands

Where stands functoriality today?

1997-01-01
R. P. Langlands
The notion of functoriality arose from the spectral analysis of automorphic forms but its definition was informed by two major theories: the theory of class fields as created by Hilbert, … The notion of functoriality arose from the spectral analysis of automorphic forms but its definition was informed by two major theories: the theory of class fields as created by Hilbert, Takagi, and Artin and others; and the representation theory of semisimple Lie groups in the form given to it by Harish-Chandra. In both theories the statements are deep and general and the proofs difficult, highly structured, and incisive. Historical antecedents and contemporary influences aside, both were in large part created by the power of one or two mathematicians. Whether for intrinsic reasons or because of the impotence of the mathematicians who have attempted to solve its problems, the fate of functoriality has been different, and the theory of automorphic forms remains in 1997 as it was in 1967: a diffuse, disordered subject driven as much by the availability of techniques as by any high esthetic purpose. Preoccupied with other matters I have drifted away from the field, so that I certainly have no remedy to offer. None the less, having had two occasions to address the question posed in the title, I have tried to understand something of the techniques that have led to progress on the questions central to functoriality, their successes and their limitations, as well as the new circumstances in the theory of automorphic forms: problems and notions that once seemed peripheral to me and whose importance I failed to appreciate are now central, either because of their intrinsic importance or because of their accessibility. Partly as an encouragement to younger, fresher mathematicians to take up the problem of functoriality, for that is one of the purposes of this school, but also partly as an idle reflection as to what I myself might undertake if I returned to it, I would like to respond to the title in broad terms, personal and certainly diffident and uncertain. My own mathematical experience and observation strongly suggest that progress is almost always the result of sustained awareness of the principal issues supplemented by some specific, concrete insight: begged, borrowed, or stolen or, happiest of all, distilled in one’s own alembic. I offer no insights. Initially there were two principal issues: functoriality itself, the relation between automorphic forms on different groups; and the identification of motivic L-functions, thus those associated to algebraic varieties over number fields, of which the zeta function, Artin Lfunctions, and the Hasse-Weil L-functions are the primitive examples, with automorphic L-functions, of which the zeta function and Hecke L-functions—of all types—are the first examples. The first issue arose and could be broached in the context of nonabelian harmonic analysis: representation theory and the trace formula. The second arose elsewhere but could

On the notion of an automorphic representation. A supplement to the preceding paper

1979-01-01
R. P. Langlands
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On Artin's L-Functions

1970-04-01
R. P. Langlands
The nonabelian Artin L-functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that of the … The nonabelian Artin L-functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that of the zeta function but little is known about their poles. It seems to be expected that the L-functions associated to irreducible representations of the Galois group or another closely related group, the Weil group, will be entire. One way that has been suggested to show this is to show that the Artin L-functions are equal to L-functions associated to automorphic forms. This is a more difficult problem than we can consider at present. In certain cases the converse question is much easier. It is possible to show that if the L-functions associated to irreducible two-dimensional representations of the Weil group are all entire then these L-functions are equal to certain L-functions associated to automorphic forms on GL(2). Before formulating this result precisely we review Weil’s generalization of Artin’s L-functions. A global field will be just an algebraic number field of finite degree over the rationals or a function field in one variable over a finite field. A local field is the completion of a global field at some place. If F is a local field let CF be the multiplicative group of F and if F is a global field let CF be the id` ele class group of F .I fK is a finite Galois extension of F the Weil group WK/F is an extension ofG(K/F ), the Galois group of K/F ,b yCK. Thus there is an exact sequence

Eisenstein series

1966-01-01
R. P. Langlands
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Shimura Varieties and the Selberg Trace Formula

1977-12-01
R. P. Langlands
This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an … This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2).
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Some contemporary problems with origins in the Jugendtraum (Hilbert’s problem 12)

1976-01-01
R. P. Langlands

SOME HOLOMORPHIC SEMI-GROUPS

1960-03-01
R. P. Langlands
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Eisenstein series

1976-01-01
R. P. Langlands

Singularités et transfert

2013-09-01
R. P. Langlands
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On Lie Semi-Groups

1960-01-01
R. P. Langlands
Suppose we have a semi-group structure defined on a subset of real Euclidean n-space, E n , by ( p, q ) → F ( p, q ) = poq. … Suppose we have a semi-group structure defined on a subset of real Euclidean n-space, E n , by ( p, q ) → F ( p, q ) = poq. In this note we shall be concerned with a representation T(.) of π as a semi-group of bounded linear operators on a Banach space 𝒳 . More particularly, we suppose that postulates P 1 , P 2 , P 3 , P 5 and P 6 of chapter 25 of (2) are satisfied so that, by Theorem 25.3.1 of that book, there is a continuous function, f(.), defined on π such that f (( ρ + σ ) a ) = f( ρa )o f ( σa ) for a ∈ π, ρ,σ ≥ 0; that the representation is strongly continuous in a neighbourhood of the origin and that T(0) = I.
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Aspects combinatoires des équations de Bethe

1997-10-17
R. P. Langlands , Yvan Saint-Aubin

Formule des Traces et Fonctorialité: le Début d'un Programme

2010-01-01
Edward Frenkel , R. P. Langlands , Ngô Bảo Châu
We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms whose eigenvalues … We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms whose eigenvalues are expressed in terms of the L-functions of automorphic representations and begin the analysis of their traces using the orbital side of the stable trace formula. We show that the most interesting part, corresponding to regular conjugacy classes, is nothing but a sum over a finite-dimensional vector space over the global field, which we call the Steinberg-Hitchin base. Therefore it may be analyzed using the Poisson summation formula. Our main result is that the leading term of the dual sum (the value at 0) is precisely the dominant term of the trace formula (the contribution of the trivial representation). This gives us hope that the full Poisson summation formula would reveal the patterns predicted by functoriality.

The Dirac monopole and induced representations

1987-01-01
R. P. Langlands
On donne un traitement mathematique du monopole de Dirac du point de vue des representations induites On donne un traitement mathematique du monopole de Dirac du point de vue des representations induites
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Eisenstein Series, the Trace Formula, and the Modern Theory of Automorphic Forms

1989-01-01
R. P. Langlands

Orbital Integrals on Forms of SL(3), II

1989-06-01
R. P. Langlands , D. Shelstad
In the paper [6] we described in a precise fashion the notion of transfer of orbital integrals from a reductive group over a local field to an endoscopic group. We … In the paper [6] we described in a precise fashion the notion of transfer of orbital integrals from a reductive group over a local field to an endoscopic group. We did not, however, prove the existence of the transfer. This remains, indeed, an unsolved problem, although in [7] we have reduced it to a local problem at the identity. In the present paper we solve this local problem for two special cases, the group SL(3), which is not so interesting, and the group SU(3), and then conclude that transfer exists for any group of type A2. The methods are those of [4], and are based on techniques of Igusa for the study of the asymptotic behavior of integrals on p-adic manifolds. (As observed in [7], the existence of the transfer over archimedean fields is a result of earlier work by Shelstad.)
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On principal values on p-adic manifolds

1983-01-01
R. P. Langlands , D. Shelstad

The Renormalization Fixed Point as a Mathematical Object

2005-01-01
R. P. Langlands

Base change for GL(2) : the theory of Saito-Shintani with applications

1975-01-01
R. P. Langlands

Dualität bei endlichen Modellen der Perkolation

1993-01-01
R. P. Langlands , Helmut Koch

Harish-Chandra, 11 October 1923 - 16 October 1983

1985-11-01
R. P. Langlands
Harish-chandra was one of the outstanding mathematicians of his generation, an algebraist and analyst, and one of those responsible for transforming infinite-dimensional group representation theory from a modest topic on … Harish-chandra was one of the outstanding mathematicians of his generation, an algebraist and analyst, and one of those responsible for transforming infinite-dimensional group representation theory from a modest topic on the periphery of mathematics and physics into a major field central to contemporary mathematics.
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Finite models for percolation

1994-01-01
R. P. Langlands , M.-A. Lafortune
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The factorization of a polynomial defined by partitions

1989-06-01
R. P. Langlands
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Un nouveau point de repère dans la théorie des formes automorphes

2007-06-01
R. P. Langlands
Ceux qui connaissent l’auteur et ses écrits, comme par exemple [L1] et [L2], savent que la notion de fonctorialité et les conjectures rattachées à celle-ci ont été introduites —en suivant … Ceux qui connaissent l’auteur et ses écrits, comme par exemple [L1] et [L2], savent que la notion de fonctorialité et les conjectures rattachées à celle-ci ont été introduites —en suivant ce que Artin avait fait pour un ensemble plus restreint de fonctions— pour aborder le problème de la prolongation analytique générale des fonctions L -automorphes. Ils savent en plus que je suis d’avis que seules les méthodes basées sur la formule des traces pourront aller au fond des problèmes. Il n’en reste pas moins que malgré de récents progrès importants sur le lemme fondamental et la formule des traces nous sommes bien loin de notre but.
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The Trace Formula and Its Applications: An Introduction to the Work of James Arthur

2001-06-01
R. P. Langlands
In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a … In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a high honour for a Canadian scientist, instituted in 1991 and awarded annually, but not previously to a mathematician, and the choice of Arthur, although certainly a recognition of his greatmerits, is also a recognition of the vigour of contemporary Canadian mathematics.
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Universality and conformal invariance for the Ising model in domains with boundary

1999-01-01
R. P. Langlands , Marc-André Lewis , Yvan Saint-Aubin
The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of conformal invariance and universality are established numerically. The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of conformal invariance and universality are established numerically.

Representation theory and arithmetic

1988-01-01
R. P. Langlands

A prologue to “Functoriality and reciprocity”, part I

2012-11-30
R. P. Langlands
Contents of Part I 1. Introduction 583 2. The local theory over the real field 593 3. The local theory over nonarchimedean fields 596 4. The global theory for algebraic … Contents of Part I 1. Introduction 583 2. The local theory over the real field 593 3. The local theory over nonarchimedean fields 596 4. The global theory for algebraic number fields 599 5. Classical algebraic number theory 602 6. Reciprocity 612 7. The geometric theory for the group GL(1) 617 8.a.The geometric theory for a general group (provisional) 641 References 661 Contents of Part II 8.b.The geometric theory for a general group (continued) 9. Gauge theory and the geometric theory 10.The p-adic theory 11.The problematics of motives
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Lectures in modern analysis and applications II

1970-01-01
R. M. Dudley , Jacob J. Feldman , Bertram Kostant +2 more

Singularités et transfert

2013-09-01
R. P. Langlands
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A prologue to “Functoriality and reciprocity”, part I

2012-11-30
R. P. Langlands
Contents of Part I 1. Introduction 583 2. The local theory over the real field 593 3. The local theory over nonarchimedean fields 596 4. The global theory for algebraic … Contents of Part I 1. Introduction 583 2. The local theory over the real field 593 3. The local theory over nonarchimedean fields 596 4. The global theory for algebraic number fields 599 5. Classical algebraic number theory 602 6. Reciprocity 612 7. The geometric theory for the group GL(1) 617 8.a.The geometric theory for a general group (provisional) 641 References 661 Contents of Part II 8.b.The geometric theory for a general group (continued) 9. Gauge theory and the geometric theory 10.The p-adic theory 11.The problematics of motives
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Formule des Traces et Fonctorialité: le Début d'un Programme

2010-01-01
Edward Frenkel , R. P. Langlands , Ngô Bảo Châu
We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms whose eigenvalues … We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms whose eigenvalues are expressed in terms of the L-functions of automorphic representations and begin the analysis of their traces using the orbital side of the stable trace formula. We show that the most interesting part, corresponding to regular conjugacy classes, is nothing but a sum over a finite-dimensional vector space over the global field, which we call the Steinberg-Hitchin base. Therefore it may be analyzed using the Poisson summation formula. Our main result is that the leading term of the dual sum (the value at 0) is precisely the dominant term of the trace formula (the contribution of the trivial representation). This gives us hope that the full Poisson summation formula would reveal the patterns predicted by functoriality.

Modular Forms and ℓ-Adic Representations

2009-02-27
R. P. Langlands

Algebro-geometric aspects of the Bethe equations

2008-04-06
R. P. Langlands , Yvan Saint-Aubin

Un nouveau point de repère dans la théorie des formes automorphes

2007-06-01
R. P. Langlands
Ceux qui connaissent l’auteur et ses écrits, comme par exemple [L1] et [L2], savent que la notion de fonctorialité et les conjectures rattachées à celle-ci ont été introduites —en suivant … Ceux qui connaissent l’auteur et ses écrits, comme par exemple [L1] et [L2], savent que la notion de fonctorialité et les conjectures rattachées à celle-ci ont été introduites —en suivant ce que Artin avait fait pour un ensemble plus restreint de fonctions— pour aborder le problème de la prolongation analytique générale des fonctions L -automorphes. Ils savent en plus que je suis d’avis que seules les méthodes basées sur la formule des traces pourront aller au fond des problèmes. Il n’en reste pas moins que malgré de récents progrès importants sur le lemme fondamental et la formule des traces nous sommes bien loin de notre but.
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Descent for Transfer Factors

2007-01-01
R. P. Langlands , D. Shelstad
In [] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do … In [] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do so in the present paper. Nonetheless we carry out what is probably an unavoidable step in any proof of existence: reduction to a local statement at the identity in the centralizer of a semisimple element, a favorite procedure of Harish Chandra that he referred to as descent.

Book Review: $p$-Adic automorphic forms on Shimura varieties

2006-10-18
R. P. Langlands
Three topics figure prominently in the modern higher arithmetic: zeta-functions, Galois representations, and automorphic forms or, equivalently, representations.The zeta-functions are attached to both the Galois representations and the automorphic representations … Three topics figure prominently in the modern higher arithmetic: zeta-functions, Galois representations, and automorphic forms or, equivalently, representations.The zeta-functions are attached to both the Galois representations and the automorphic representations and are the link that joins them.Although by and large abstruse and often highly technical, the subject has many claims on the attention of mathematicians as a whole: the spectacular solution of Fermat's Last Theorem; concrete conjectures that are both difficult and not completely inaccessible, above all that of Birch and Swinnerton-Dyer; roots in an ancient tradition of the study of algebraic irrationalities; a majestic conceptual architecture with implications not confined to number theory; and great current vigor.Nevertheless, in spite of major results modern arithmetic remains inchoate, with far more conjectures than theorems.There is no schematic introduction to it that reveals the structure of the conjectures whose proofs are its principal goal and of the methods to be employed, and for good reason.There are still too many uncertainties.I nonetheless found while preparing this review that without forming some notion of the outlines of the final theory, I was quite at sea with the subject and with the book.So ill-equipped as I am in many ways -although not in all -my first, indeed my major, task was to take bearings.The second is, bearings taken, doubtful or not, to communicate them at least to an experienced reader and, so far as this is possible, even to an inexperienced one.For lack of time and competence I accomplished neither task satisfactorily.So, although I have made a real effort, this review is not the brief, limpid yet comprehensive, account of the subject, revealing its manifold possibilities, that I would have liked to write and that it deserves.The review is imbalanced and there is too much that I had to leave obscure, too many possibly premature intimations.A reviewer with greater competence who saw the domain whole and, in addition, had a command of the detail would have done much better. 1 It is perhaps best to speak of L-functions rather than of zeta-functions and to begin not with p-adic functions but with those that are complex-valued and thusat least in principle, although one problem with which the theory is confronted is to establish this in general -analytic functions in the whole complex plane with only a very few poles.The Weil zeta-function of a smooth algebraic variety over a finite field is a combinatorial object defined by the number of points on the variety over the field itself and its Galois extensions.The Hasse-Weil zeta-function of a smooth variety over a number field F is the product over all places p of the zeta-function of the variety reduced at p. Of course, the reduced variety may not be smooth for some p and for those some additional care has to be taken with the definition.In fact it is not the Hasse-Weil zeta-function itself which is of greatest interest, but rather factors of its numerator and denominator, especially, but not necessarily, the 2000 Mathematics Subject Classification.Primary 11G18, 14G10. 1 These lines will mean more to the reader who consults the supplement to the review that is posted with it on the site http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html.
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The Renormalization Fixed Point as a Mathematical Object

2005-01-01
R. P. Langlands

The Trace Formula and Its Applications: An Introduction to the Work of James Arthur

2001-06-01
R. P. Langlands
In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a … In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a high honour for a Canadian scientist, instituted in 1991 and awarded annually, but not previously to a mathematician, and the choice of Arthur, although certainly a recognition of his greatmerits, is also a recognition of the vigour of contemporary Canadian mathematics.
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Universality and conformal invariance for the Ising model in domains with boundary

1999-01-01
R. P. Langlands , Marc-André Lewis , Yvan Saint-Aubin
The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of conformal invariance and universality are established numerically. The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of conformal invariance and universality are established numerically.

Representations of abelian algebraic groups

1997-12-01
R. P. Langlands
There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive … There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension
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Aspects combinatoires des équations de Bethe

1997-10-17
R. P. Langlands , Yvan Saint-Aubin

Where stands functoriality today?

1997-01-01
R. P. Langlands
The notion of functoriality arose from the spectral analysis of automorphic forms but its definition was informed by two major theories: the theory of class fields as created by Hilbert, … The notion of functoriality arose from the spectral analysis of automorphic forms but its definition was informed by two major theories: the theory of class fields as created by Hilbert, Takagi, and Artin and others; and the representation theory of semisimple Lie groups in the form given to it by Harish-Chandra. In both theories the statements are deep and general and the proofs difficult, highly structured, and incisive. Historical antecedents and contemporary influences aside, both were in large part created by the power of one or two mathematicians. Whether for intrinsic reasons or because of the impotence of the mathematicians who have attempted to solve its problems, the fate of functoriality has been different, and the theory of automorphic forms remains in 1997 as it was in 1967: a diffuse, disordered subject driven as much by the availability of techniques as by any high esthetic purpose. Preoccupied with other matters I have drifted away from the field, so that I certainly have no remedy to offer. None the less, having had two occasions to address the question posed in the title, I have tried to understand something of the techniques that have led to progress on the questions central to functoriality, their successes and their limitations, as well as the new circumstances in the theory of automorphic forms: problems and notions that once seemed peripheral to me and whose importance I failed to appreciate are now central, either because of their intrinsic importance or because of their accessibility. Partly as an encouragement to younger, fresher mathematicians to take up the problem of functoriality, for that is one of the purposes of this school, but also partly as an idle reflection as to what I myself might undertake if I returned to it, I would like to respond to the title in broad terms, personal and certainly diffident and uncertain. My own mathematical experience and observation strongly suggest that progress is almost always the result of sustained awareness of the principal issues supplemented by some specific, concrete insight: begged, borrowed, or stolen or, happiest of all, distilled in one’s own alembic. I offer no insights. Initially there were two principal issues: functoriality itself, the relation between automorphic forms on different groups; and the identification of motivic L-functions, thus those associated to algebraic varieties over number fields, of which the zeta function, Artin Lfunctions, and the Hasse-Weil L-functions are the primitive examples, with automorphic L-functions, of which the zeta function and Hecke L-functions—of all types—are the first examples. The first issue arose and could be broached in the context of nonabelian harmonic analysis: representation theory and the trace formula. The second arose elsewhere but could

A Combinatorial Laplacian with Vertex Weights

1996-08-01
Fan Chung , R. P. Langlands
One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of … One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see [1, 17, 11, 15]). The usual notion of the combinatorial Laplacian for a graph involves edge weights. Namely, a Laplacian L forGis a matrix with rows and columns indexed by the vertex setVofG, and the (u,v)-entry of L, foru,vinG,u≠v, is associated with the edge-weight of the edge (u,v). It is not so obvious to consider Laplacians with vertex weights (except for using some symmetric combinations of the vertex weights to define edge-weights). In this note, we consider a vertex weighted Laplacian which is motivated by a problem arising in the study of algebro-geometric aspects of the Bethe Ansatz [18]. The usual Laplacian can be regarded as a special case with all vertex-weights equal. We will generalize the matrix-tree theorem to matrix-tree theorems of counting "rooted" directed spanning trees. In addition, the characteristic polynomial of the vertex-weighted Laplacian has coefficients with similar interpretations. We also consider subgraphs with non-trial boundary. We will shown that the Laplacian with Dirichlet boundary condition has its determinant equal to the number of rooted spanning forests. The usual matrix-tree theorem is a special case with the boundary consisting of one single vertex.

Book Review: Elliptic curves

1994-01-01
R. P. Langlands
REVIEWStheory of fiber bundles and lead naturally to the curvature definition of the characteristic classes.3. Exterior differential systems include any system of partial differential equations.The theory is intrinsic and adapts … REVIEWStheory of fiber bundles and lead naturally to the curvature definition of the characteristic classes.3. Exterior differential systems include any system of partial differential equations.The theory is intrinsic and adapts well to nonlinear equations.This list could continue.Exterior differential calculus is destined to occupy a more important place in multivariable calculus!Of which works was Carian most proud?Through my conversations with him during my Paris séjour of 1936-1937, I would like to venture a guess.I would suggest that it is his works on linear representations, including his discovery of spinors in 1913.He published a book on spinors in 1938 in which he included the physical applications.In the conclusion of the book the authors state: "As a rule, Cartan built his scientific research on works of his predecessors, developing their ideas so well that other mathematicians often forgot the original works."This clearly does not apply to the works mentioned above.Moreover, in the examples used by the authors to illustrate their statement-moving frames and generalized spaces-Cartan developed the methods and ideas for homogeneous spaces with any Lie group, which went far beyond the scope of his predecessors.Cartan roamed through a vast and fertile area of mathematics.With his power and insight he was able to pick up the gems wherever he treaded.His books are full of interesting details.This was not the case with his earlier works, such as pseudogroups and exterior differential systems, which were original but needed clarification.As a result the recognition of his achievements came late.It was perhaps Hermann Weyl's work on group representations in 1925-1926 that made Cartan famous in the general mathematical community.Although Poincaré had a high opinion of the role of a "group" in mathematics and of Cartan's contribution to it (his report on Cartan's work is included in the book), Cartan was elected a member of the French Academy only in 1931.1 I have no doubt that Cartan realized the importance of his works.He was able to ignore the outside reaction and led a simple, happy, and fruitful life.
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Conformal invariance in two-dimensional percolation

1994-01-01
R. P. Langlands , Philippe Pouliot , Yvan Saint-Aubin
The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it … The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the seventeenth century. It was introduced more recently into mathematics by S. R. Broadbent and J. M. Hammersley ([BH]) and is a branch of probability theory that is especially close to statistical mechanics. Broadbent and Hammersley distinguish between two types of spreading of a fluid through a medium, or between two aspects of the probabilistic models of such processes: diffusion processes, in which the random mechanism is ascribed to the fluid; and percolation processes, in which it is ascribed to the medium. A percolation process typically depends on one or more probabilistic parameters. For example, if molecules of a gas are absorbed at the surface of a porous solid (as in a gas mask) then their ability to penetrate the solid depends on the sizes of the pores in it and their positions, both conceived to be distributed in some random manner. A simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner (that could be determined by a periodic graph), and to be open (thus very large) or closed (thus smaller than the molecules) with probabilities p and 1 − p. As p increases the probability of deeper penetration of the gas into the interior of the solid grows.
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Conformal invariance in two-dimensional percolation

1994-01-01
R. P. Langlands , Philippe Pouliot , Yvan Saint-Aubin
The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique … The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It was rather to describe as concretely as possible, although in hypothetical form, the geometric aspects of universality, especially conformal invariance, in the context of percolation, and to present the numerical results that support the hypotheses. On the other hand, one ulterior purpose is to draw the attention of mathematicians to the mathematical problems posed by the physical notions. Some precise basic definitions are necessary simply to orient the reader. Moreover a brief description of scaling and universality on the one hand and of renormalization on the other is also essential in order to establish their physical importance and to clarify their mathematical content.

Finite models for percolation

1994-01-01
R. P. Langlands , M.-A. Lafortune
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Conformal invariance in two-dimensional percolation

1994-01-01
R. P. Langlands , Philippe Pouliot , Yvan Saint-Aubin
The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique … The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It was rather to describe as concretely as possible, although in hypothetical form, the geometric aspects of universality, especially conformal invariance, in the context of percolation, and to present the numerical results that support the hypotheses. On the other hand, one ulterior purpose is to draw the attention of mathematicians to the mathematical problems posed by the physical notions. Some precise basic definitions are necessary simply to orient the reader. Moreover a brief description of scaling and universality on the one hand and of renormalization on the other is also essential in order to establish their physical importance and to clarify their mathematical content.
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Dualität bei endlichen Modellen der Perkolation

1993-01-01
R. P. Langlands , Helmut Koch

On the universality of crossing probabilities in two-dimensional percolation

1992-05-01
R. P. Langlands , C. Pichet , Ph. Pouliot +1 more
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The factorization of a polynomial defined by partitions

1989-06-01
R. P. Langlands
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Orbital Integrals on Forms of SL(3), II

1989-06-01
R. P. Langlands , D. Shelstad
In the paper [6] we described in a precise fashion the notion of transfer of orbital integrals from a reductive group over a local field to an endoscopic group. We … In the paper [6] we described in a precise fashion the notion of transfer of orbital integrals from a reductive group over a local field to an endoscopic group. We did not, however, prove the existence of the transfer. This remains, indeed, an unsolved problem, although in [7] we have reduced it to a local problem at the identity. In the present paper we solve this local problem for two special cases, the group SL(3), which is not so interesting, and the group SU(3), and then conclude that transfer exists for any group of type A2. The methods are those of [4], and are based on techniques of Igusa for the study of the asymptotic behavior of integrals on p-adic manifolds. (As observed in [7], the existence of the transfer over archimedean fields is a result of earlier work by Shelstad.)
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Eisenstein Series, the Trace Formula, and the Modern Theory of Automorphic Forms

1989-01-01
R. P. Langlands

On the classification of irreducible representations of real algebraic groups

1989-01-01
R. P. Langlands
Suppose G is a connected reductive group over a global field F . Many of the problems of the theory of automorphic forms involve some aspect of study of the … Suppose G is a connected reductive group over a global field F . Many of the problems of the theory of automorphic forms involve some aspect of study of the representation ρ of G(A(F )) on the space of slowly increasing functions on the homogeneous space G(F )\G(A(F )). It is of particular interest to study the irreducible constituents of ρ. In a lecture [9], published some time ago, but unfortunately rendered difficult to read by a number of small errors and a general imprecision, reflections in part of a hastiness for which my excitement at the time may be to blame, I formulated some questions about these constituents which seemed to me then, as they do today, of some fascination. The questions have analogues when F is a local field; these concern the irreducible admissible representations of G(F ). As I remarked in the lecture, there are cases in which the answers to the questions are implicit in existing theories. If G is abelian they are consequences of class field theory, especially of the Tate-Nakayama duality. This is verified in [10]. If F is the real or complex field, they are consequences of the results obtained by Harish-Chandra for representations ∗Preprint, Institute for Advanced Study, 1973. Appeared in Math. Surveys and Monographs, No. 31, AMS (1988)

Representation theory and arithmetic

1988-01-01
R. P. Langlands

On the definition of transfer factors

1987-03-01
R. P. Langlands , D. Shelstad

The Dirac monopole and induced representations

1987-01-01
R. P. Langlands
On donne un traitement mathematique du monopole de Dirac du point de vue des representations induites On donne un traitement mathematique du monopole de Dirac du point de vue des representations induites
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Harish-Chandra, 11 October 1923 - 16 October 1983

1985-11-01
R. P. Langlands
Harish-chandra was one of the outstanding mathematicians of his generation, an algebraist and analyst, and one of those responsible for transforming infinite-dimensional group representation theory from a modest topic on … Harish-chandra was one of the outstanding mathematicians of his generation, an algebraist and analyst, and one of those responsible for transforming infinite-dimensional group representation theory from a modest topic on the periphery of mathematics and physics into a major field central to contemporary mathematics.
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Orbital Integrals on Forms of SL(3), I

1983-04-01
R. P. Langlands

On principal values on p-adic manifolds

1983-01-01
R. P. Langlands , D. Shelstad

BASE CHANGE FOR GL(2)

1980-01-01
R. P. Langlands
R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a … R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic representation of the group GL(2, /A), /A being the adele ring of the field, and L(s, ?), whose definition is ultimately due to Hecke, is known to be entire. The main result, from which the existence of ? follows, is that it is always possible to transfer automorphic representations of GL(2) over one number field to representations over a cyclic extension of the field. The tools he employs here are the trace formula and harmonic analysis on the group GL(2) over a local field.

On the Zeta-Functions of Some Simple Shimura Varieties

1979-12-01
R. P. Langlands
In an earlier paper [ 14 ] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real … In an earlier paper [ 14 ] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [ 13 ]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [ 13 ] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [ 17 ].
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<i>L</i>-Indistinguishability For <i>SL</i> (2)

1979-08-01
J.-P. Labesse , R. P. Langlands
The notion of L -indistinguishability, like many others current in the study of L -functions, has yet to be completely defined, but it is in our opinion important for the … The notion of L -indistinguishability, like many others current in the study of L -functions, has yet to be completely defined, but it is in our opinion important for the study of automorphic forms and of representations of algebraic groups. In this paper we study it for the simplest class of groups, basically forms of SL (2). Although the definition we use is applicable to very few groups, there is every reason to believe that the results will have general analogues [ 12 ]. The phenomena which the notion is intended to express have been met–and exploited–by others (Hecke [ 5 ] § 13, Shimura [ 17 ]). Their source seems to lie in the distinction between conjugacy and stable conjugacy. If F is a field, G a reductive algebraic group over F, and the algebraic closure of F then two elements of G(F) may be conjugate in without being conjugate in G(F) .
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Stable Conjugacy: Definitions and Lemmas

1979-08-01
R. P. Langlands
The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and … The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious. The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.
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On the notion of an automorphic representation. A supplement to the preceding paper

1979-01-01
R. P. Langlands
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Automorphic representations, Shimura varieties, and motives. Ein Märchen

1979-01-01
R. P. Langlands
1. Introduction. It had been my intention to survey the problems posed by the study of zetafunctions of Shimura varieties. But I was too sanguine. This would be a mammoth … 1. Introduction. It had been my intention to survey the problems posed by the study of zetafunctions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representationtheoretic, but they both arise in the study of the zeta-function at the infinite places. The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too imprecise, and were not even in principle amenable to proof by Shimura’s methods of descent. The conjecture as it stands is the only statement I could discover that meets his criticism and is compatible with Shimura’s conjecture. The statement of the conjecture must be preceded by some constructions, which have implications that had escaped me. When combined with Deligne’s conception of Shimura varieties as parameter varieties for families of motives they suggest the introduction of a group, here called the Taniyama group, which may be of importance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed. With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of formulating one or two specific conjectures, but we begin to weave a tissue of surmise and hypothesis, and curiosity drives us on. Deligne’s ideas are reviewed in the fourth section, but to understand them one must be familiar at least with the elements of the formalism of tannakian categories underlying the conjectural theory of motives, say, with the main results of Chapter II of [40]. The present Summer Institute is predicated on the belief that there is a close relation between automorphic representations and motives. The relation is usually couched in terms of L-functions, * First appeared in Automorphic forms, representations, and L-functions, Proc. of Symp. in Pure

Shimura Varieties and the Selberg Trace Formula

1977-12-01
R. P. Langlands
This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an … This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2).
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On the Functional Equations Satisfied by Eisenstein Series

1976-01-01
R. P. Langlands
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Some functional equations

1976-01-01
R. P. Langlands

Cusp forms

1976-01-01
R. P. Langlands

Miscellaneous lemmas

1976-01-01
R. P. Langlands

Eisenstein series

1976-01-01
R. P. Langlands

Some contemporary problems with origins in the Jugendtraum (Hilbert’s problem 12)

1976-01-01
R. P. Langlands

Base change for GL(2) : the theory of Saito-Shintani with applications

1975-01-01
R. P. Langlands

On Artin's L-Functions

1970-04-01
R. P. Langlands
The nonabelian Artin L-functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that of the … The nonabelian Artin L-functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that of the zeta function but little is known about their poles. It seems to be expected that the L-functions associated to irreducible representations of the Galois group or another closely related group, the Weil group, will be entire. One way that has been suggested to show this is to show that the Artin L-functions are equal to L-functions associated to automorphic forms. This is a more difficult problem than we can consider at present. In certain cases the converse question is much easier. It is possible to show that if the L-functions associated to irreducible two-dimensional representations of the Weil group are all entire then these L-functions are equal to certain L-functions associated to automorphic forms on GL(2). Before formulating this result precisely we review Weil’s generalization of Artin’s L-functions. A global field will be just an algebraic number field of finite degree over the rationals or a function field in one variable over a finite field. A local field is the completion of a global field at some place. If F is a local field let CF be the multiplicative group of F and if F is a global field let CF be the id` ele class group of F .I fK is a finite Galois extension of F the Weil group WK/F is an extension ofG(K/F ), the Galois group of K/F ,b yCK. Thus there is an exact sequence

Problems in the theory of automorphic forms to Salomon Bochner in gratitude

1970-01-01
R. P. Langlands

Automorphic Forms on GL (2)

1970-01-01
H. Jacquet , R. P. Langlands
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Lectures in modern analysis and applications II

1970-01-01
R. M. Dudley , Jacob J. Feldman , Bertram Kostant +2 more
Coauthor Papers Together
Yvan Saint-Aubin 7
D. Shelstad 4
H. Jacquet 2
Philippe Pouliot 2
Edward Frenkel 1
B. Kostant 1
J. Feldman 1
E. M. Stein 1
Ph. Pouliot 1
Ngô Bảo Châu 1
R. M. Dudley 1
C. Pichet 1
Philippe Pouliot 1
R. M. Dudley 1
Jacob J. Feldman 1
J.-P. Labesse 1
E. M. Stein 1
Fan Chung 1
Helmut Koch 1
Bertram Kostant 1
Marc-André Lewis 1
M.-A. Lafortune 1

Automorphic Forms on GL (2)

1970-01-01
H. Jacquet , R. P. Langlands
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<i>L</i>-Indistinguishability For <i>SL</i> (2)

1979-08-01
J.-P. Labesse , R. P. Langlands
The notion of L -indistinguishability, like many others current in the study of L -functions, has yet to be completely defined, but it is in our opinion important for the … The notion of L -indistinguishability, like many others current in the study of L -functions, has yet to be completely defined, but it is in our opinion important for the study of automorphic forms and of representations of algebraic groups. In this paper we study it for the simplest class of groups, basically forms of SL (2). Although the definition we use is applicable to very few groups, there is every reason to believe that the results will have general analogues [ 12 ]. The phenomena which the notion is intended to express have been met–and exploited–by others (Hecke [ 5 ] § 13, Shimura [ 17 ]). Their source seems to lie in the distinction between conjugacy and stable conjugacy. If F is a field, G a reductive algebraic group over F, and the algebraic closure of F then two elements of G(F) may be conjugate in without being conjugate in G(F) .
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On the universality of crossing probabilities in two-dimensional percolation

1992-05-01
R. P. Langlands , C. Pichet , Ph. Pouliot +1 more
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Percolation Theory for Mathematicians.

1984-01-01
J. M. Hammersley , Harry Kesten
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Orbital Integrals on Forms of SL(3), I

1983-04-01
R. P. Langlands

Regular elements of semisimple algebraic groups

1965-12-01
Robert Steinberg
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Rational conjugacy classes in reductive groups

1982-12-01
Robert Kottwitz

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem

1961-09-01
Bertram Kostant

Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories

1986-10-01
John Cardy

Harmonic Analysis on Real Reductive Groups

1977-01-01
V. S. Varadarajan
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BASE CHANGE FOR GL(2)

1980-01-01
R. P. Langlands
R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a … R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic representation of the group GL(2, /A), /A being the adele ring of the field, and L(s, ?), whose definition is ultimately due to Hecke, is known to be entire. The main result, from which the existence of ? follows, is that it is always possible to transfer automorphic representations of GL(2) over one number field to representations over a cyclic extension of the field. The tools he employs here are the trace formula and harmonic analysis on the group GL(2) over a local field.

Characters and inner forms of a quasi-split group over $R$

1979-01-01
D. Shelstad

Harmonic Analysis on Semi-Simple Lie Groups II

1972-01-01
Garth Warner

A Theorem on Semi-Simple P-adic Groups

1972-03-01
J. A. Shalika
Let k be a non-archimedean local field and G a connected, semi-simple algebraic group defined over k. If G = G(k) denotes the group of k-rational points of G, then … Let k be a non-archimedean local field and G a connected, semi-simple algebraic group defined over k. If G = G(k) denotes the group of k-rational points of G, then G, with its natural topology, is locally compact. Let G' and V denote respectively the sets of regular and unipotent elements of G, and let Cc(G) denote the space of locally constant, complex valued functions on G having compact support. For x E G' U V, let G(x) denote the conjugacy class of G containing x. G(x) carries an essentially unique G-invariant measure Pa (? 1.2). Fix a normalization of pa. For f E Cc(G) and x as above, let

Hecke Polynomials as Congruence ζ Functions in Elliptic Modular Case

1967-03-01
Yasutaka Ihara
Introduction. Notation. ? 1. Preliminaries on quadratic fields. (Prop. 1 & Corol. 1, 2. Def. 1) ? 2. Preliminaries on elliptic curves. (Th. D, Prop. 2, 2', 3, 4, 4', … Introduction. Notation. ? 1. Preliminaries on quadratic fields. (Prop. 1 & Corol. 1, 2. Def. 1) ? 2. Preliminaries on elliptic curves. (Th. D, Prop. 2, 2', 3, 4, 4', 5. Lemma 1, 2. Def. 2) ? 3. C functions of fibre varieties. (Th. 1, Lemma 3,4. Def. 3) ?4. Hecke polynomials and their expressions by the Zr(u). (Th. 2, Lemma 5, 6. Def. 4) ? 5. The verification of Lemma 6. (Lemma 7, 8, 9) Supplement and References.

On Artin's L-Functions

1970-04-01
R. P. Langlands
The nonabelian Artin L-functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that of the … The nonabelian Artin L-functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that of the zeta function but little is known about their poles. It seems to be expected that the L-functions associated to irreducible representations of the Galois group or another closely related group, the Weil group, will be entire. One way that has been suggested to show this is to show that the Artin L-functions are equal to L-functions associated to automorphic forms. This is a more difficult problem than we can consider at present. In certain cases the converse question is much easier. It is possible to show that if the L-functions associated to irreducible two-dimensional representations of the Weil group are all entire then these L-functions are equal to certain L-functions associated to automorphic forms on GL(2). Before formulating this result precisely we review Weil’s generalization of Artin’s L-functions. A global field will be just an algebraic number field of finite degree over the rationals or a function field in one variable over a finite field. A local field is the completion of a global field at some place. If F is a local field let CF be the multiplicative group of F and if F is a global field let CF be the id` ele class group of F .I fK is a finite Galois extension of F the Weil group WK/F is an extension ofG(K/F ), the Galois group of K/F ,b yCK. Thus there is an exact sequence

On Discontinuous Groups Operating on the Product of the Upper Half Planes

1963-01-01
Hideo Shimizu
Let H be the direct product of the n upper half planes, and let be the connected component of the identity of the group of all analytic automorphisms of Hn. … Let H be the direct product of the n upper half planes, and let be the connected component of the identity of the group of all analytic automorphisms of Hn. is the direct product of n subgroups G1, G2, . . .* Gq each of which is isomorphic to the group of all analytic automorphisms of the upper half plane H. Consider a discrete subgroup of such that the factor space 17\G is of finite measure. (The notations Hn, or will keep, unless otherwise stated, these meanings throughout this paper.) For the study of the groups of this type, it is important to investigate the case where is an irreducible subgroup of in the following sense. A discrete subgroup of is said to be irreducible if is not commensurable' with any direct product F' x F, where F' and F are respectively discrete subgroups of the partial products G' and G of =G, x G2 x ... x Ga with = G' x G, G' I {1}, G # {1}. The main purpose of this paper was originally to calculate the dimension of the space of cusp foruis for an irreducible group by means of Selberg's trace formula; however, for the sake of completeness, and in view of the fact that no proof has been published as yet for the results stated by Pyatetzki-Shapiro in [6], it has been found desirable to prove these results here, following the ideas indicated by Pyatetzki-Shapiro himself. Therefore, no new results will be found in our ?? 1-3 except for some supplementary results such as Theorem 1, and for Theorems 6 and 7, which are proved under an additional condition on the fundamental domain of (Assumption (F) in No. 11, ?3). Theorems 2, 3, 4, 5 are the restatements of the results of Pyatetzki-Shapiro; but, for the sake of simplicity, we state here the latter three theorems only for irreducible case. For these results, Pyatetzki-Shapiro has indicated in [6] only a sketch of proof for Theorem 3 as well as the implications between them. However, our proofs will be probably more or less the same as the ones he has. In ??4-5, we shall calculate the dimension of the space of cusp forms for an irreducible group under the assumption (F). The main result is

On the definition of transfer factors

1987-03-01
R. P. Langlands , D. Shelstad

The Selberg Trace Formula for Groups of F-rank One

1974-09-01
James Arthur

On Zeta Functions of Quaternion Algebras

1965-01-01
Hideo Shimizu
The present paper is concerned with some equalities between zeta functions of quaternion algebras introduced in Godement [6], Shimura [11], Tamagawa [13]. Let A be a quaternion algebra over a … The present paper is concerned with some equalities between zeta functions of quaternion algebras introduced in Godement [6], Shimura [11], Tamagawa [13]. Let A be a quaternion algebra over a totally real algebraic number field 1? of degree m, and let D be an order in A; let S be the idele group of A, and U the group of units in S with respect to D; let p be a representation of ( a/f)*, f being an integral two-sided rD-ideal, and let ki (1 < i < m) be non-negative integers. These being given, we can speak of a space of automorphic functions associated with (p, {ki}) (cf. ? 2.2 and ? 2.5) and of a repre-sentation Z of the Hecke ring 9R(U, @) in this space. Let T(q) be the sum of all integral elements in 9R(U, @ of norm q and let C(s) be the Dirichlet. series defined by

Dimension of spaces of automorphic forms

1966-01-01
R. P. Langlands
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Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series

1956-09-01
Atle Selberg
In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these … In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these investigations is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions). This relation we here refer to as the trace-formula.

Algebraic number theory

1999-11-22
Kazuya Katô , Nobushige Kurokawa , Takeshi Saito
This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background … This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.
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Kroneckerian Model of Fields of Elliptic Modular Functions

1959-07-01
Jun-ichi Igusa

On the Algebraic Curves Uniformized by Arithmetical Automorphic Functions

1967-11-01
Koji Doi , Hidehisa Naganuma
The purpose of this note is to prove a theorem on algebraic curves uniformized by automorphic functions with respect to a discontinuous group obtained from an indefinite quaternion algebra, and … The purpose of this note is to prove a theorem on algebraic curves uniformized by automorphic functions with respect to a discontinuous group obtained from an indefinite quaternion algebra, and to study some related problems. The theorem may be viewed as an addition to a recent work of Shimura [8] (quoted hereafter as [C]). To state it we repeat here the notation of [C]. Let B be a quaternion algebra over a totally real algebraic number field F of finite degree. We assume that B is unramified at the real archimedean prime of F corresponding to the identity mapping of F, and ramified at other archimedean primes of F. Here we consider F as a subfield of the real number field R. We denote by D(B/F) the product of all prime ideals of F which are ramified in B. Let B+ denote the set of all elements a of B such that NB!F(a) is totally positive. Then the elements of B+ act naturally on the upper half complex plane &. For every maximal order o in B and an integral ideal c in F, we put

Automorphic representations of unitary groups in three variables

1990-01-01
Jonathan David Rogawski
The purpose of this book is to develop the stable trace formula for unitary groups in three variables. The stable trace formula is then applied to obtain a classification of … The purpose of this book is to develop the stable trace formula for unitary groups in three variables. The stable trace formula is then applied to obtain a classification of automorphic representations. This work represents the first case in which the stable trace formula has been worked out beyond the case of SL (2) and related groups. Many phenomena which will appear in the general case present themselves already for these unitary groups.

�ber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichlet scher Reihen durch Funktionalgleichungen

1949-12-01
Hans Maaß

Stable Conjugacy: Definitions and Lemmas

1979-08-01
R. P. Langlands
The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and … The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious. The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.
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Discrete series for semisimple Lie groups. II: Explicit determination of the characters

1966-01-01
Harish-chandra Harish-Chandra
Let G be a connected semisimple Lie group and K a maximal compact subgroup of G.We shall show in this paper that G has a discrete series (see [4 (d), … Let G be a connected semisimple Lie group and K a maximal compact subgroup of G.We shall show in this paper that G has a discrete series (see [4 (d), w 5]) ff and only if it has a compact Cartan subgroup B. Let Ea denote the set of all equivalence classes of irreducible unitary representations of G, wMch are square-integrable.For any ~o E ~a, let | denote the character, X~ the infinitesimal character and d(r the formal degree (see [4 (d), w 3]) of co.Then it is known [4 (d), w 5] that the distribution
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Critical percolation in finite geometries

1992-02-21
John Cardy
The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are … The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.
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The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups

1966-01-01
R. P. Langlands

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem

2009-01-01
Bertram Kostant

Representations of abelian algebraic groups

1997-12-01
R. P. Langlands
There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive … There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension
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L-Indistinguishability for real groups

1982-09-01
D. Shelstad

Stable trace formula: Elliptic singular terms

1986-09-01
Robert Kottwitz

On some results of Atkin and Lehner

1973-12-01
W. Casselman

Shalika’s germs for<i>p</i>-adic GL(<i>n</i>). II. The subregular term

1984-07-01
Joe Repka
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The Dimension of Spaces of Automorphic Forms

1963-01-01
R. P. Langlands
1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the … 1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in iarish-Chandra's investigations of the representations of semi-simple groups. These integrals have been evaluated. In this paper, after clarifying the relation between the two types of integrals, we go on to complete the evaluation of the integrals appearing in the trace formula. Before the formula for the dimension that results is described let us review iarish-Chandra's construction of bounded symmetric domains and introduce the automorphic forms to be considered. If (7 is the connected component of the identity in the group of pseudoconformal mappings of a bounded symmetric domain then (7 has a trivial centre and a maximal compact subgroup of any simple component has nondiscrete centre. Conversely if (7 is a connected semi-simple group with these two properties then (7 is the connected component of the identity in the group of pseudo-conformal mappings of a bounded symmetric domain [2(d)]. Let g be the Lie algebra of (7 and gc its complexification. Let GC be the simplyconnected complex Lie group with Lie algebra gc; replace G by the connected subgroup G of GC with Lie algebra g. Let K be a maximal compact sub
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Eisensteinkohomologie und die Konstruktion gemischter Motive

1993-01-01
Günter Harder
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On some problems suggested by the trace formula

1983-01-01
James Arthur

Infinite conformal symmetry in two-dimensional quantum field theory

1984-07-01
A. A. Belavin , A. Polyakov , A. B. Zamolodchikov
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On Dirichlet Series and Abelian Varieties Attached to Automorphic Forms

1962-09-01
Goro Shimura
Since Hecke [13] had given a general theory of constructing Dirichlet series with Euler-product and functional equation out of elliptic modular forms of any level, several authors considered its generalization … Since Hecke [13] had given a general theory of constructing Dirichlet series with Euler-product and functional equation out of elliptic modular forms of any level, several authors considered its generalization for other types of automorphic forms. In the case of the Hilbert modular group of level one, Herrmann [14] succeeded in this problem; he has shown the necessity of considering not only the product of the upper half-planes but also the domain r consisting of the points (zl, * * , Zr) of the r-dimensional complex vector space Cr such that Im(z,) # 0, * ., Im(zr) # 0, and h distinct discontinuous groups commensurable to each other, h being the class number of the totally real number field in the problem. On the other hand, the unit-group of an order in an indefinite quaternion algebra over the rational number field Q yields a fuchsian group. In this case, Eichler [6] defined Hecke's operators as representations of algebraic correspondences, called modular correspondences, and proved a formula for the trace of the operators. The trace-formula was proved also by Selberg [22] in a more general formulation. Recently, Godement [9] has given a theory of zeta-functions attached to division algebras; namely, he has shown the possibility of applying the adele-idele method of Iwasawa-Tate [15, 28] to automorphic functions and forms with respect to the unit-group of a division algebra over Q. The case of non-holomorphic automorphic functions of this type has been investigated by Tamagawa [27]. In [25] I have treated cusp-forms with respect to the unit-group of an indefinite quaternion algebra over Q. Now, the purpose of Part I of the present paper is to develop an analogous theory for automorphic forms on the domain ; mentioned above with respect to the unit-groups of an indefinite quaternion algebra. Let F be a totally real algebraic number field of degree t, and D be a quaternion algebra over F. Denote by r the number of infinite prime spots of F unramified in D, and suppose that r > 0. Let D,1 and D-2 denote the product of r copies of the total matric algebra M2(R) over the real number field R and the product of t r copies of the division ring K of real quaternions, respectively. Then, D( Q R is isomorphic to D,1 x D12. Let o be a maximal order in D and I' be the group of units in o. Let y, and 72 237

Algebro-geometric aspects of the Bethe equations

2008-04-06
R. P. Langlands , Yvan Saint-Aubin

Convolution of L² - functions

1968-01-01
Neil W. Rickert
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Characters of irreducible representations of the virasoro algebra

1984-03-01
Alvany Rocha-Caridi , Nolan R. Wallach

Dualität bei endlichen Modellen der Perkolation

1993-01-01
R. P. Langlands , Helmut Koch

Graph theory and statistical physics

1971-05-01
J W Essam

On the classification of irreducible representations of real algebraic groups

1989-01-01
R. P. Langlands
Suppose G is a connected reductive group over a global field F . Many of the problems of the theory of automorphic forms involve some aspect of study of the … Suppose G is a connected reductive group over a global field F . Many of the problems of the theory of automorphic forms involve some aspect of study of the representation ρ of G(A(F )) on the space of slowly increasing functions on the homogeneous space G(F )\G(A(F )). It is of particular interest to study the irreducible constituents of ρ. In a lecture [9], published some time ago, but unfortunately rendered difficult to read by a number of small errors and a general imprecision, reflections in part of a hastiness for which my excitement at the time may be to blame, I formulated some questions about these constituents which seemed to me then, as they do today, of some fascination. The questions have analogues when F is a local field; these concern the irreducible admissible representations of G(F ). As I remarked in the lecture, there are cases in which the answers to the questions are implicit in existing theories. If G is abelian they are consequences of class field theory, especially of the Tate-Nakayama duality. This is verified in [10]. If F is the real or complex field, they are consequences of the results obtained by Harish-Chandra for representations ∗Preprint, Institute for Advanced Study, 1973. Appeared in Math. Surveys and Monographs, No. 31, AMS (1988)

Sharpness of the phase transition in percolation models

1987-09-01
Michael Aizenman , David J. Barsky
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Stable trace formula: Cuspidal tempered terms

1984-09-01
Robert Kottwitz
Consider a connected reductive group G over a number field F. For technical reasons we assume that the derived group of G is simply connected (see [L1]). in [L3] Langlands … Consider a connected reductive group G over a number field F. For technical reasons we assume that the derived group of G is simply connected (see [L1]). in [L3] Langlands partially stabilizes the trace formula for G. After making certain assumptions, he writes the elliptic regular part of the trace formula for G as a linear combination of the elliptic G-regular parts of the stable trace formulas for the elliptic endoscopic groups H of G. The function f/ used in the stable trace formula forH is obtained from the function f used in the trace formula for G by transferring orbital integrals.
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