704 pages 1 5/16 spine

James ArthurDavid EllwoodRobert KottwitzEditors

Clay Mathematics ProceedingsVolume 4

American Mathematical Society

Clay Mathematics Institute

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Proceedings of the Clay Mathematics Institute2003 Summer School, The Fields Institute Toronto, Canada, June 227, 2003

The modern theory of automorphic forms, embodied inwhat has come to be known as the Langlands program,is an extraordinary unifying force in mathematics. Itproposes fundamental relations that tie arithmeticinformation from number theory and algebraic geometrywith analytic information from harmonic analysis andgroup representations. These reciprocity laws,conjectured by Langlands, are still largely unproved.However, their capacity to unite large areas ofmathematics insures that they will be a central area of study for years to come.

The goal of this volume is to provide an entry point intothis exciting and challenging field. It is directed on theone hand at graduate students and professionalmathematicians who would like to work in the area.The longer articles in particular represent an attempt to enable a reader to master some of the more difficulttechniques. On the other hand, the book will also beuseful to mathematicians who would like simply tounderstand something of the subject. They will be ableto consult the expository portions of the various articles.

The volume is centered around the trace formula andShimura varieties. These areas are at the heart of thesubject, but they have been especially difficult to learnbecause of a lack of expository material. The volumeaims to rectify the problem. It is based on the coursesgiven at the 2003 Clay Mathematics Institute SummerSchool. However, many of the articles have beenexpanded into comprehensive introductions, either tothe trace formula or the theory of Shimura varieties, orto some aspect of the interplay and application of thetwo areas.

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HARMONIC ANALYSIS,THE TRACE FORMULA, AND SHIMURA VARIETIES

HARMONIC ANALYSIS,THE TRACE FORMULA, AND SHIMURA VARIETIESProceedings of the Clay Mathematics Institute2003 Summer School, The Fields InstituteToronto, Canada, June 227, 2003

James ArthurDavid EllwoodRobert KottwitzEditors

Clay Mathematics ProceedingsVolume 4

American Mathematical Society

Clay Mathematics Institute

2000 Mathematics Subject Classification. Primary 1102; Secondary 11F70, 11F72,11F85, 11G18, 14G35, 22E35, 22E50, 22E55.

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10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05

Contents

Preface vii

An Introduction to the Trace Formula 1James Arthur

Introduction to Shimura Varieties 265J. S. Milne

Linear Algebraic Groups 379Fiona Murnaghan

Harmonic Analysis on Reductive p-adic Groups and Lie Algebras 393Robert E. Kottwitz

Homogeneity for Reductive p-adic Groups: An Introduction 523Stephen DeBacker

Compactifications and Cohomology of Modular Varieties 551Mark Goresky

Introduction to Shimura Varieties with Bad Reduction of Parahoric Type 583Thomas J. Haines

A Statement of the Fundamental Lemma 643Thomas C. Hales

Notes on the Generalized Ramanujan Conjectures 659Peter Sarnak

List of Participants 687

v

Preface

This volume is based on lectures given at the fourth Clay Mathematics Insti-tute Summer School entitled Harmonic Analysis, the Trace Formula, and ShimuraVarieties. It was held at the Fields Institute in Toronto, Canada, from June 2 toJune 27, 2003.

The main goal of the School was to introduce graduate students and youngmathematicians to three broad and interrelated areas in the theory of automorphicforms. Much of the volume is comprised of the articles of Arthur, Kottwitz, andMilne. Although these articles are based on lectures given at the school, the authorshave chosen to go well beyond what was discussed there, in order to provide both asense of the underlying structure of the subject and a working knowledge of someof its techniques. They were written to be self-contained in some places, and tobe used in conjunction with given references in others. We hope the volume willconvey the depth and beauty of this challenging field, in which there yet remainsso much to be discoveredperhaps some of it by you, the reader!

The theory of automorphic forms is formulated in terms of reductive algebraicgroups. This is sometimes a serious obstacle for mathematicians whose backgrounddoes not include Lie groups and Lie algebras. The monograph is by no means in-tended to exclude such mathematicians, even though the theory of reductive groupswas an informal prerequisite for the Summer School. Some modest familiarity withthe language of algebraic groups is often sufficient, at least to get started. Forthis reason, we have generally resisted the temptation to work with specific matrixgroups. The short article of Murnaghan contains a summary of some of the basicproperties of reductive algebraic groups that are used elsewhere in the monograph.

Much of the modern theory of automorphic forms is governed by two funda-mental problems that are at the heart of the Langlands program. One is Lang-lands principle of functoriality. The other is the general analogue of the Shimura-Taniyama-Weil conjecture on modular elliptic curves. (See [A] and [L, 2].) Theseproblems are among the deepest questions in mathematics. It is premature to tryto guess what various techniques will play a role in their ultimate resolution. How-ever, the trace formula and the theory of Shimura varieties are both likely to bean essential part of the story. They have already been used to establish significantspecial cases.

The trace formula has perhaps been more closely identified with the first prob-lem. Special cases of functoriality arise naturally from the conjectural theory ofendoscopy, in which a comparison of trace formulas would be used to characterizethe internal structure of the automorphic representations of a given group. (See[Sh] for a discussion of the first case to be investigated.) Likewise, Shimura varietiesare usually associated with the second problem. As higher dimensional analoguesof modular curves, they are attached by definition to certain reductive groups. Inmany cases, it has been possible to establish reciprocity laws between -adic Ga-lois representations on their cohomology groups and automorphic representationsof the corresponding reductive groups. These laws can be formulated as an ex-plicit formula for the zeta function of a Shimura variety in terms of automorphicL-functions. (See [K] for a discussion of the rough form such a formula is expectedto take. The word rough should be taken seriously, given the current limitationsof our understanding.)

vii

viii PREFACE

The work of Wiles that led to a proof of Fermats Last Theorem suggests thatthe two problems are inextricably linked. This is already apparent in the reciprocitylaws that have been established for Shimura varieties. Indeed, the conjectural for-mula for the zeta function of a general Shimura variety requires the theory ofendoscopy even to state. Moreover, the proof of these reciprocity laws requiresa comparison of the (automorphic) trace formula with an (-adic) Lefschetz traceformula. Some of the most striking parts of the argument are in the comparisonof the various terms in the two formulas. The most sophisticated Shimura vari-eties for which there are complete results are the so-called Picard modular surfaces.(See [LR], especially the summary on pp. 255302.) Picard modular surfaces areattached to unitary groups in three variables. It is no coincidence that the the-ory of endoscopy has also been established for these groups, thereby yielding aclassification of their automorphic representations [R].

There is some discussion of these problems in the articles of Arthur and Milne.However, the articles of both Arthur and Milne really are intended as introductions,despite their length. The theory of endoscopy, and the automorphic description ofzeta functions of Shimura varieties, are at the forefront of present day research.They are for the most part beyond the scope of this monograph.

The local terms in the trace formula are essentially analytic objects. Theyinclude the invariant orbital integrals and irreducible characters that are the basisfor Harish-Chandras theory of local harmonic analysis. They also include weightedorbital integrals and weighted characters, objects that arose for the first time withthe trace formula. The article of Kottwitz is devoted to the general study of theseterms at p-adic places. It is a largely self-contained course, which covers manyof Harish-Chandras basic results in invariant harmonic analysis, as well as theirweighted, noninvariant analogues.

The article of DeBacker focuses on the phenomenon of homogeneity in invari-ant harmonic analysis at p-adic places. It concerns quantitative forms of some ofthe basic theorems of p-adic harmonic analysis, such as Howes finiteness theoremand Harish-Chandras local character expansion. The article also explains how ho-mogeneity enters into Waldspurgers analysis of stability for linear combinations ofnilpotent orbital integrals.

There are subtle questions concerning the terms in the trace formula that gobeyond those treated by Kottwitz and DeBacker. The most basic of these is knownas the fundamental lemma, even though it is still largely conjectural.1 The article byHales contains a precise statement of the conjecture and some remarks on progresstoward a general proof. The fundamental lemma occupies a unique place in thetheory. It is a critical ingredient in the comparison of trace formulas that is partof the theory of endoscopy. It has an equally indispensable role in the comparisonof (automorphic and -adic) trace formulas needed to establish reciprocity laws forShimura varieties.

Some Shimura varieties are projective, which is to say that they are compact ascomplex varieties. They correspond to reductive groups over Q that are anisotropic.The trace formula in this case simplifies considerably. It reduces to the Selberg traceformula for compact quotient. On the other hand, the arithmetic geometry of suchvarieties is still very rich. In particular, the comparison of individual terms in the

1Moreover, the term lemma is ultimately a gross understatement.

PREFACE ix

two kinds of trace formulas is of major interest. There is a great deal left to bedone, but it is in this case that there has been the most progress.

If the Shimura variety is not projective, the comparison is more sophisticated.It has to be based on the relationship between L2-cohomology and intersection coho-mology, conjectured by Zucker, and established by Saper and Stern, and Looijenga.The article of Goresky describes several compactifications of open Shimura vari-eties and their relations with associated cohomology groups. Goreskys article alsoserves as an introduction to work of Goresky and MacPherson, in which weightedcohomology complexes on the reductive Borel-Serre compactification are used toobtain a Lefschetz formula for the intersection cohomology of the Baily-Borel com-pactification. According to Zuckers conjecture, this last formula is equivalent tothe relevant form of the automorphic trace formula. There remains the importantopen problem of establishing a corresponding -adic Lefschetz formula that can becompared with either one of these two formulas.

The reciprocity laws proved for Picard modular surfaces in [LR] apply to placesof good reduction. The same restriction has been implicit in our discussion of otherShimura varieties. In the final analysis, one would like to establish reciprocitylaws between -adic Galois representations and automorphic representations thatapply to all places. The theory of Shimura varieties at places of bad reduction isconsiderably less developed, although there has certainly been progress. The articleof Haines is a survey of recent work in this direction, concentrating on the case oflevel structures of parahoric type. It also touches upon the problem of comparingthe automorphic trace formula with the Lefschetz formula, now in the context ofbad reduction.

The article of Sarnak concerns the classical Ramanujan conjecture for modularforms and its higher dimensional analogues. Langlands has shown that the gen-eralized Ramanujan conjecture is a consequence of the principle of functoriality.Conversely, it is possible that the generalized Ramanujan conjecture could play acritical role in the study of those cases of functoriality that are not part of the the-ory of endoscopy. Sarnak describes the present state of the conjecture and discussesvarious techniques that have been successfully applied to special cases.

We have tried to present the contents of the monograph from a unified perspec-tive. Our description has been centered around two fundamental problems that arethe essential expression of the Langlands program. The two problems ought tobe treated as signposts, which give direction to current work, but which point todestinations that will not be reached in the foreseeable future. The reader is free todraw whatever inspiration from them his or her temperament permits. In any case,many of the questions discussed in the various articles here are of great interestin their own right. In point of fact, there is probably too much in the monographfor anyone to learn in a limited period of time. Perhaps the best strategy for abeginner would be to start with one or two articles of special interest, and try tomaster them.

As we have mentioned, participants were encouraged to bring a prior under-standing of the basic properties of algebraic groups. The theory of reductive groupsis rooted in the structure of complex semisimple Lie algebras, for which [Se] and [H]are good references. As for algebraic groups themselves, a familiarity with many ofthe topics in [B] or [Sp] is certainly desirable, though perhaps not essential.

x PREFACE

Participants were also assumed to have some knowledge of number theory. Themain theorem of class field theory is reviewed without proof in the article of Milne.A complete treatment can be found in [CF]. Tates article on global class fieldtheory in this reference contains a particularly good introduction to the theory.The thesis of Tate, reprinted as a separate article in [CF], is also recommendedfor its introduction to adeles and its construction of the basic abelian automorphicL-functions.

A reader might also want to consult other general articles in automorphic forms.A good introductory reference to the general theory of automorphic forms is theproceedings of the Edinburgh instructional conference [BK].

This Clay Mathematics Institute Summer School could not have taken placewithout the efforts of many people. We deeply appreciate the role of the ClayMathematics Institute in making this summer school possible, and thank VidaSalahi in particular for the care and attention she exercised in bringing the volumeto its final form. We are most grateful to the staff of the Fields Institute, who didsuch a superb job of making the School run smoothly. We are equally indebtedto all the lecturers, not only for agreeing to take part in the School, but also forproviding the texts collected in this volume. Last, but surely not least, we wouldlike to thank the participants, whose enthusiastic response made it all worthwhile.

James Arthur, David Ellwood, Robert Kottwitz.August, 2005.

References

[A] J. Arthur, The principle of functoriality, Bull. Amer. Math. Soc. 40 (2002), 3953

[BK] T.N. Baily and A.W. Knapp, Representation Theory and Automorphic Forms, Proc.Sympos. Pure Math. 61, Amer. Math. Soc., 1996.

[B] A. Borel, Linear Algebraic Groups, Benjamin, 1969.[CF] J. Cassels and A. Frohlich, Algebraic Number Theory, Thompson, 1967.

[H] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag,1972.

[K] R. Kottwitz, Shimura varieties and -adic representations, in Automorphic Forms,

Shimura Varieties, and L-functions, vol. I, Academic Press, 1990, 161209.[L] R. Langlands, Automorphic representations, Shimura varieties, and motives. Ein

Marchen, in Automorphic Forms, Representations and L-functions, Proc. Sympos. PureMath. vol. 33, Part 2, Amer. Math. Soc., 1979, 205246.

[LR] R. Langlands and D. Ramakrishnan, The Zeta Functions of Picard Modular Surfaces,Les Publications CRM, Montreal, 1992.

[R] J. Rogawski, Automorphic Representations of Unitary Groups in Three Variables, Ann.of Math. Studies 123, Princeton Univ. Press, 1990.

[Se] J.-P. Serre, Complex Semisimple Lie Algebras, Springer-Verlag, 1987.[Sh] D. Shelstad, Notes on L-indistinguishability (based on a lecture by R.P. Langlands), in

Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. vol.33, Part 2, Amer. Math. Soc., 1979, 185192.

[Sp] T. Springer, Linear Algebraic Groups, Birkhauser, 1981.

PREFACE xi

Summer School Lecture Courses: June 220, 2003

Introduction to the trace formulaJ. Arthur, June 220

Introduction to Shimura varietiesJ. Milne, June 220

Background from algebraic groupsF. Murnaghan, June 26

Harmonic analysis on reductive groups and Lie algebrasR. Kottwitz, June 920

Advanced Short Courses: June 2327, 2003

An introduction to homogeneity with applicationsS. DeBacker

Geometry and topology of compactifications of modular varietiesM. Goresky

Bad reduction of Shimura varietiesT. Haines

An introduction to the fundamental lemmaT. Hales

Analytic aspects of automorphic formsP. Sarnak

Clay Mathematics ProceedingsVolume 4, 2005

An Introduction to the Trace Formula

James Arthur

Contents

Foreword 3

Part I. The Unrefined Trace Formula 71. The Selberg trace formula for compact quotient 72. Algebraic groups and adeles 113. Simple examples 154. Noncompact quotient and parabolic subgroups 205. Roots and weights 246. Statement and discussion of a theorem 297. Eisenstein series 318. On the proof of the theorem 379. Qualitative behaviour of JT (f) 46

10. The coarse geometric expansion 5311. Weighted orbital integrals 5612. Cuspidal automorphic data 6413. A truncation operator 6814. The coarse spectral expansion 7415. Weighted characters 81

Part II. Refinements and Applications 8916. The first problem of refinement 8917. (G,M)-families 9318. Local behaviour of weighted orbital integrals 10219. The fine geometric expansion 10920. Application of a Paley-Wiener theorem 11621. The fine spectral expansion 12622. The problem of invariance 13923. The invariant trace formula 14524. A closed formula for the traces of Hecke operators 15725. Inner forms of GL(n) 166

Supported in part by NSERC Discovery Grant A3483.

c 2005 Clay Mathematics Institute

1

2 JAMES ARTHUR

26. Functoriality and base change for GL(n) 18027. The problem of stability 19228. Local spectral transfer and normalization 20429. The stable trace formula 21630. Representations of classical groups 234

Afterword: beyond endoscopy 251

References 258

Foreword

These notes are an attempt to provide an entry into a subject that has notbeen very accessible. The problems of exposition are twofold. It is important topresent motivation and background for the kind of problems that the trace formulais designed to solve. However, it is also important to provide the means for acquiringsome of the basic techniques of the subject. I have tried to steer a middle coursebetween these two sometimes divergent objectives. The reader should refer to earlierarticles [Lab2], [Lan14], and the monographs [Sho], [Ge], for different treatmentsof some of the topics in these notes.

I had originally intended to write fifteen sections, corresponding roughly tofifteen lectures on the trace formula given at the Summer School. These sectionscomprise what has become Part I of the notes. They include much introductorymaterial, and culminate in what we have called the coarse (or unrefined) trace for-mula. The coarse trace formula applies to a general connected, reductive algebraicgroup. However, its terms are too crude to be of much use as they stand.

Part II contains fifteen more sections. It has two purposes. One is to transformthe trace formula of Part I into a refined formula, capable of yielding interestinginformation about automorphic representations. The other is to discuss some ofthe applications of the refined formula. The sections of Part II are considerablylonger and more advanced. I hope that a familiarity with the concepts of Part Iwill allow a reader to deal with the more difficult topics in Part II. In fact, the latersections still include some introductory material. For example, 16, 22, and 27contain heuristic discussions of three general problems, each of which requires afurther refinement of the trace formula. Section 26 contains a general introductionto Langlands principle of functoriality, to which many of the applications of thetrace formula are directed.

We begin with a discussion of some constructions that are part of the founda-tions of the subject. In 1 we review the Selberg trace formula for compact quotient.In 2 we introduce the ring A = AF of adeles. We also try to illustrate why adelicalgebraic groups G(A), and their quotients G(F )\G(A), are more concrete objectsthan they might appear at first sight. Section 3 is devoted to examples related to1 and 2. It includes a brief description of the Jacquet-Langlands correspondencebetween quaternion algebras and GL(2). This correspondence is a striking exampleof the kind of application of which the trace formula is capable. It also illustratesthe need for a trace formula for noncompact quotient.

In 4, we begin the study of noncompact quotient. We work with a generalalgebraic group G, since this was a prerequisite for the Summer School. However,we have tried to proceed gently, giving illustrations of a number of basic notions.For example, 5 contains a discussion of roots and weights, and the related objectsneeded for the study of noncompact quotient. To lend Part I an added appearanceof simplicity, we work over the ground field Q, instead of a general number field F .

The rest of Part I is devoted to the general theme of truncation. The problem isto modify divergent integrals so that they converge. At the risk of oversimplifying

3

4 JAMES ARTHUR

matters, we have tried to center the techniques of Part I around one basic result,Theorem 6.1. Corollary 10.1 and Theorem 11.1, for example, are direct corollariesof Theorem 6.1, as well as essential steps in the overall construction. Other resultsin Part I also depend in an essential way on either the statement of Theorem 6.1or a key aspect of its proof. Theorem 6.1 itself asserts that a truncation of thefunction

K(x, x) =

G(Q)f(x1x), f Cc

(G(A)

),

is integrable. It is the integral of this function over G(Q)\G(A) that yields a traceformula in the case of compact quotient. The integral of its truncation in the generalcase is what leads eventually to the coarse trace formula at the end of Part I.

After stating Theorem 6.1 in 6, we summarize the steps required to convertthe truncated integral into some semblance of a trace formula. We sketch the proofof Theorem 6.1 in 8. The arguments here, as well as in the rest of Part I, areboth geometric and combinatorial. We present them at varying levels of generality.However, with the notable exception of the review of Eisenstein series in 7, we havetried in all cases to give some feeling for what is the essential idea. For example,we often illustrate geometric points with simple diagrams, usually for the specialcase G = SL(3). The geometry for SL(3) is simple enough to visualize, but oftencomplicated enough to capture the essential point in a general argument. I amindebted to Bill Casselman, and his flair for computer graphics, for the diagrams.The combinatorial arguments are used in conjunction with the geometric argumentsto eliminate divergent terms from truncated functions. They rely ultimately on thatsimplest of cancellation laws, the binomial identity

FS

(1)|F | ={

0, if S = ,1, if S = ,

which holds for any finite set S (Identity 6.2).The parallel sections 11 and 15 from the later stages of Part I anticipate the

general discussion of 1621 in Part II. They provide refined formulas for genericterms in the coarse trace formula. These formulas are explicit expressions, whoselocal dependence on the given test function f is relatively transparent. The firstproblem of refinement is to establish similar formulas for all of the terms. Becausethe remaining terms are indexed by conjugacy classes and representations that aresingular, this problem is more difficult than any encountered in Part I. The solutionrequires new analytic techniques, both local and global. It also requires extensionsof the combinatorial techniques of Part I, which are formulated in 17 as propertiesof (G,M)-families. We refer the reader to 1621 for descriptions of the variousresults, as well as fairly substantial portions of their proofs.

The solution of the first problem yields a refined trace formula. We summarizethis new formula in 22, in order to examine why it is still not satisfactory. Theproblem here is that its terms are not invariant under conjugation of f by elementsin G(A). They are in consequence not determined by the values taken by f atirreducible characters. We describe the solution of this second problem in 23. Ityields an invariant trace formula, which we derive by modifying the terms in therefined, noninvariant trace formula so that they become invariant in f .

FOREWORD 5

In 2426 we pause to give three applications of the invariant trace formula.They are, respectively, a finite closed formula for the traces of Hecke operators oncertain spaces, a term by term comparison of invariant trace formulas for generallinear groups and central simple algebras, and cyclic base change of prime order forGL(n). It is our discussion of base change that provides the opportunity to reviewLanglands principle of functoriality.

The comparisons of invariant trace formulas in 25 and 26 are directed atspecial cases of functoriality. To study more general cases of functoriality, onerequires a third refinement of the trace formula.

The remaining problem is that the terms of the invariant trace formula are notstable as linear forms in f . Stability is a subtler notion than invariance, and ispart of Langlands conjectural theory of endoscopy. We review it in 27. In 28and 29 we describe the last of our three refinements. This gives rise to a stabletrace formula, each of whose terms is stable in f . Taken together, the results of29 can be regarded as a stabilization process, by which the invariant trace formulais decomposed into a stable trace formula, and an error term composed of stabletrace formulas on smaller groups. The results are conditional upon the fundamentallemma. The proofs, conditional as they may be, are still too difficult to permit morethan passing comment in 29.

The general theory of endoscopy includes a significant number of cases of func-toriality. However, its avowed purpose is somewhat different. The principal aim ofthe theory is to analyze the internal structure of representations of a given group.Our last application is a broad illustration of what can be expected. In 30 wedescribe a classification of representations of quasisplit classical groups, both localand global, into packets. These results depend on the stable trace formula, andthe fundamental lemma in particular. They also presuppose an extension of thestabilization of 29 to twisted groups.

As a means for investigating the general principle of functoriality, the theoryof endoscopy has very definite limitations. We have devoted a word after 30 tosome recent ideas of Langlands. The ideas are speculative, but they seem also torepresent the best hope for attacking the general problem. They entail using thetrace formula in ways that are completely new.

These notes are really somewhat of an experiment. The style varies from sectionto section, ranging between the technical and the discursive. The more difficulttopics typically come in later sections. However, the progression is not alwayslinear, or even monotonic. For example, the material in 1315, 1921, 23, and25 is no doubt harder than much of the broader discussion in 16, 22, 26, and27. The last few sections of Part II tend to be more discursive, but they are alsohighly compressed. This is the price we have had to pay for trying to get close tothe frontiers. The reader should feel free to bypass the more demanding passages,at least initially, in order to develop an overall sense of the subject.

It would not have been possible to go very far by insisting on complete proofs.On the other hand, a survey of the results might have left a reader no closerto acquiring any of the basic techniques. The compromise has been to includesomething representative of as many arguments as possible. It might be a sketch ofthe general proof, a suggestive proof of some special case, or a geometric illustrationby a diagram. For obvious reasons, the usual heading PROOF does not appearin the notes. However, each stated result is eventually followed by a small box

6 JAMES ARTHUR

, when the discussion that passes for a proof has come to an end. This ought tomake the structure of each section more transparent. My hope is that a determinedreader will be able to learn the subject by reinforcing the partial arguments here,when necessary, with the complete proofs in the given references.

Part I. The Unrefined Trace Formula

1. The Selberg trace formula for compact quotient

Suppose that H is a locally compact, unimodular topological group, and that is a discrete subgroup of H. The space \H of right cosets has a right H-invariantBorel measure. Let R be the unitary representation of H by right translation onthe corresponding Hilbert space L2(\H). Thus,

(R(y)

)(x) = (xy), L2(\H), x, y H.

It is a fundamental problem to decompose R explicitly into irreducible unitaryrepresentations. This should be regarded as a theoretical guidepost rather than aconcrete goal, since one does not expect an explicit solution in general. In fact,even to state the problem precisely requires the theory of direct integrals.

The problem has an obvious meaning when the decomposition of R is discrete.Suppose for example that H is the additive group R, and that is the subgroupof integers. The irreducible unitary representations of R are the one dimensionalcharacters x ex, where ranges over the imaginary axis iR. The representationR decomposes as direct sum over such characters, as ranges over the subset 2iZof iR. More precisely, let R be the unitary representation of R on L2(Z) defined by

(R(y)c

)(n) = e2inyc(n), c L2(Z).

The correspondence that maps L2(Z\R) to its set of Fourier coefficients

(n) =

Z\R(x)e2inxdx, n Z,

is then a unitary isomorphism from L2(Z\R) onto L2(Z), which intertwines therepresentations R and R. This is of course the Plancherel theorem for Fourierseries.

The other basic example to keep in mind occurs where H = R and = {1}.In this case the decomposition of R is continuous, and is given by the Planchereltheorem for Fourier transforms. The general intuition that can inform us is asfollows. For arbitrary H and , there will be some parts of R that decomposediscretely, and therefore behave qualitatively like the theory of Fourier series, andothers that decompose continuously, and behave qualitatively like the theory ofFourier transforms.

In the general case, we can study R by integrating it against a test functionf Cc(H). That is, we form the operator

R(f) =

H

f(y)R(y)dy

7

8 JAMES ARTHUR

on L2(\H). We obtain

(R(f)

)(x) =

H

(f(y)R(y)

)(x)dy

=

H

f(y)(xy)dy

=

H

f(x1y)(y)dy

=

\H

(

f(x1y))(y)dy,

for any L2(\H) and x H. It follows that R(f) is an integral operator withkernel

(1.1) K(x, y) =

f(x1y), x, y \H.

The sum over is finite for any x and y, since it may be taken over the intersectionof the discrete group with the compact subset

x supp(f)y1

of H.For the rest of the section, we consider the special case that \H is compact.

The operator R(f) then acquires two properties that allow us to investigate itfurther. The first is that R decomposes discretely into irreducible representations, with finite multiplicities m(,R). This is not hard to deduce from the spectraltheorem for compact operators. Since the kernel K(x, y) is a continuous function onthe compact space (\H)(\H), and is hence square integrable, the correspondingoperator R(f) is of Hilbert-Schmidt class. One applies the spectral theorem to thecompact self adjoint operators attached to functions of the form

f(x) = (g g)(x) =

H

g(y)g(x1y)dy, g Cc(H).

The second property is that for many functions, the operator R(f) is actually oftrace class, with

(1.2) trR(f) =

\HK(x, x)dx.

If H is a Lie group, for example, one can require that f be smooth as well ascompactly supported. Then R(f) becomes an integral operator with smooth kernelon the compact manifold \H. It is well known that (1.2) holds for such operators.

Suppose that f is such that (1.2) holds. Let {} be a set of representatives ofconjugacy classes in . For any and any subset of H, we write for the

1. THE SELBERG TRACE FORMULA FOR COMPACT QUOTIENT 9

centralizer of in . We can then write

tr(R(f)

)=

\H

K(x, x)dx

=

\H

f(x1x)dx

=

\H

{}

\

f(x11x)dx

=

{}

\H

f(x1x)dx

=

{}

H\H

\H

f(x1u1ux)du dx

=

{}vol(\H)

H\H

f(x1x)dx.

These manipulations follow from Fubinis theorem, and the fact that for any se-quence H1 H2 H of unimodular groups, a right invariant measure on H1\Hcan be written as the product of right invariant measures on H2\H and H1\H2respectively. We have obtained what may be regarded as a geometric expansionof tr

(R(f)

)in terms of conjugacy classes in . By restricting R(f) to the irre-

ducible subspaces of L2(\H), we obtain a spectral expansion of R(f) in terms ofirreducible unitary representations of H.

The two expansions tr(R(f)

)provide an identity of linear forms

(1.3)

aH ()fH() =

aH ()fH(),

where is summed over (a set of representatives of) conjugacy classes in , and is summed over (equivalence classes of) irreducible unitary representatives of H.The linear forms on the geometric side are invariant orbital integrals

(1.4) fH() =

H\Hf(x1x)dx,

with coefficientsaH () = vol(\H),

while the linear forms on the spectral side are irreducible characters

(1.5) fH() = tr((f)

)= tr

(H

f(y)(y)dy),

with coefficientsaH () = m(,R).

This is the Selberg trace formula for compact quotient.We note that if H = R and = Z, the trace formula (1.3) reduces to the

Poisson summation formula. For another example, we could take H to be a finitegroup and f(x) to be the character tr(x) of an irreducible representation of H.In this case, (1.3) reduces to a special case of the Frobenius reciprocity theorem,which applies to the trivial one dimensional representation of the subgroup of H.(A minor extension of (1.3) specializes to the general form of Frobenius reciprocity.)

10 JAMES ARTHUR

Some of Selbergs most striking applications of (1.3) were to the group H =SL(2,R) of real, (22)-matrices of determinant one. Suppose that X is a compactRiemann surface of genus greater than 1. The universal covering surface of Xis then the upper half plane, which we identify as usual with the space of cosetsSL(2,R)/SO(2,R). (Recall that the compact orthogonal group K = SO(2,R) isthe stabilizer of

1 under the transitive action of SL(2,R) on the upper half

plane by linear fractional transformations.) The Riemann surface becomes a spaceof double cosets

X = \H/K,

where is the fundamental group of X, embedding in SL(2,R) as a discrete sub-group with compact quotient. By choosing left and right K-invariant functionsf Cc (H), Selberg was able to apply (1.3) to both the geometry and analysis ofX.

For example, closed geodesics on X are easily seen to be bijective with conju-gacy classes in . Given a large positive integer N , Selberg chose f so that the lefthand side of (1.3) approximated the number g(N) of closed geodesics of length lessthan N . An analysis of the corresponding right hand side gave him an asymptoticformula for g(N), with a sharp error term. Another example concerns the Laplace-Beltrami operator attached to X. In this case, Selberg chose f so that the righthand side of (1.3) approximated the number h(N) of eigenvalues of less than N .An analysis of the corresponding left hand side then provided a sharp asymptoticestimate for h(N).

The best known discrete subgroup of H = SL(2,R) is the group = SL(2,Z)of unimodular integral matrices. In this case, the quotient \H is not compact.The example of = SL(2,Z) is of special significance because it comes with thesupplementary operators introduced by Hecke. Hecke operators include a family ofcommuting operators {Tp} on L2(\H), parametrized by prime numbers p, whichcommute also with the action of the group H = SL(2,R). The families {cp}of simultaneous eigenvalues of Hecke operators on L2(\H) are known to be offundamental arithmetic significance. Selberg was able to extend his trace formula(1.3) to this example, and indeed to many other quotients of rank 1. He alsoincluded traces of Hecke operators in his formulation. In particular, he obtained afinite closed formula for the trace of Tp on any space of classical modular forms.

Selberg worked directly with Riemann surfaces and more general locally sym-metric spaces, so the role of group theory in his papers is less explicit. We canrefer the reader to the basic articles [Sel1] and [Sel2]. However, many of Selbergsresults remain unpublished. The later articles [DL] and [JL, 16] used the languageof group theory to formulate and extend Selbergs results for the upper half plane.

In the next section, we shall see how to incorporate the theory of Hecke oper-ators into the general framework of (1.1). The connection is through adele groups,where Hecke operators arise in a most natural way. Our ultimate goal is to describea general trace formula that applies to any adele group. The modern role of sucha trace formula has changed somewhat from the original focus of Selberg. Ratherthan studying geometric and spectral data attached to a given group in isolation,one tries to compare such data for different groups. In particular, one would liketo establish reciprocity laws among the fundamental arithmetic data associated toHecke operators on different groups.

2. ALGEBRAIC GROUPS AND ADELES 11

2. Algebraic groups and adeles

Suppose that G is a connected reductive algebraic group over a number fieldF . For example, we could take G to be the multiplicative group GL(n) of invertible(n n)-matrices, and F to be the rational field Q. Our interest is in the generalsetting of the last section, with equal to G(F ). It is easy to imagine that thisgroup could have arithmetic significance. However, it might not be at all clearhow to embed discretely into a locally compact group H. To do so, we have tointroduce the adele ring of F .

Suppose for simplicity that F equals the rational field Q. We have the usualabsolute value v() = | | on Q, and its corresponding completion Qv = Q =R. For each prime number p, there is also a p-adic absolute value vp() = | |p onQ, defined by

|t|p = pr, t = prab1,for integers r, a and b with (a, p) = (b, p) = 1. One constructs its completionQvp = Qp by a process identical to that of R. As a matter of fact, | |p satisfies anenhanced form of the triangle inequality

|t1 + t2|p max{|t1|p, |t2|p

}, t1, t2 Q.

This has the effect of giving the compact unit ball

Zp ={tp Qp : |tp|p 1

}in Qp the structure of a subring of Qp. The completions Qv are all locally compactfields. However, there are infinitely many of them, so their direct product is notlocally compact. One forms instead the restricted direct product

A =restv

Qv = Rrestp

Qp = R Afin

={t = (tv) : tp = tvp Zp for almost all p

}.

Endowed with the natural direct limit topology, A = AQ becomes a locally compactring, called the adele ring of Q. The diagonal image of Q in A is easily seen to bediscrete. It follows that H = G(A) is a locally compact group, in which = G(Q)embeds as a discrete subgroup. (See [Tam2].)

A similar construction applies to a general number field F , and gives rise to alocally compact ring AF . The diagonal embedding

= G(F ) G(AF ) = Hexhibits G(F ) as a discrete subgroup of the locally compact group G(AF ). However,we may as well continue to assume that F = Q. This represents no loss of generality,since one can pass from F to Q by restriction of scalars. To be precise, if G1 isthe algebraic group over Q obtained by restriction of scalars from F to Q, then = G(F ) = G1(Q), and H = G(AF ) = G1(A).

We can define an automorphic representation of G(A) informally to be anirreducible representation of G(A) that occurs in the decomposition of R. Thisdefinition is not precise for the reason mentioned in 1, namely that there could bea part of R that decomposes continuously. The formal definition [Lan6] is in factquite broad. It includes not only irreducible unitary representations of G(A) in thecontinuous spectrum, but also analytic continuations of such representations.

12 JAMES ARTHUR

The introduction of adele groups appears to have imposed a new and perhapsunwelcome level of abstraction onto the subject. The appearance is illusory. Sup-pose for example that G is a simple group over Q. There are two possibilities:either G(R) is noncompact (as in the case G = SL(2)), or it is not. If G(R) isnoncompact, the adelic theory for G may be reduced to the study of of arithmeticquotients of G(R). As in the case G = SL(2) discussed at the end of 1, this isclosely related to the theory of Laplace-Beltrami operators on locally symmetricRiemannian spaces attached to G(R). If G(R) is compact, the adelic theory re-duces to the study of arithmetic quotients of a p-adic group G(Qp). This in turn isclosely related to the spectral theory of combinatorial Laplace operators on locallysymmetric hypergraphs attached to the Bruhat-Tits building of G(Qp).

These remarks are consequences of the theorem of strong approximation. Sup-pose that S is a finite set of valuations of Q that contains the archimedean valuationv. For any G, the product

G(QS) =vS

G(Qv)

is a locally compact group. Let KS be an open compact subgroup of G(AS), where

AS ={t A : tv = 0, v S

}is the ring theoretic complement of QS in A. Then G(FS)KS is an open subgroupof G(A).

Theorem 2.1. (a) (Strong approximation) Suppose that G is simply connected,in the sense that the topological space G(C) is simply connected, and that G(QS)is noncompact for every simple factor G of G over Q. Then

G(A) = G(Q) G(QS)KS .(b) Assume only that G(QS) is noncompact for every simple quotient G of G

over Q. Then the set of double cosets

G(Q)\G(A)/G(QS)KS

is finite.

For a proof of (a) in the special case G = SL(2) and S = {v}, see [Shim,Lemma 6.15]. The reader might then refer to [Kne] for a sketch of the generalargument, and to [P] for a comprehensive treatment. Part (b) is essentially acorollary of (a).

According to (b), we can write G(A) as a disjoint union

G(A) =n

i=1

G(Q) xi G(QS)KS ,

for elements x1 = 1, x2, . . . , xn in G(AS). We can therefore write

G(Q)\G(A)/KS =n

i=1

(G(Q)\G(Q) xi G(QS)KS/KS

)=

ni=1

(iS\G(QS)

),

2. ALGEBRAIC GROUPS AND ADELES 13

for discrete subgroups

iS = G(QS) (G(Q) xiKS(xi)1

)of G(QS). We obtain a G(QS)-isomorphism of Hilbert spaces

(2.1) L2(G(Q)\G(A)/KS

) = ni=1

L2(iS\G(QS)

).

The action of G(QS) on the two spaces on each side of (2.1) is of course by righttranslation. It corresponds to the action by right convolution on either space byfunctions in the algebra Cc

(G(QS)

). There is a supplementary convolution algebra,

the Hecke algebra H(G(AS),KS

)of compactly supported functions on G(AS) that

are left and right invariant under translation by KS . This algebra acts by rightconvolution on the left hand side of (2.1), in a way that clearly commutes with theaction of G(QS). The corresponding action ofH

(G(AS),KS

)on the right hand side

of (2.1) includes general analogues of the operators defined by Hecke on classicalmodular forms.

This becomes more concrete if S = {v}. Then AS equals the subring Afin ={t A : t = 0} of finite adeles in A. If G satisfies the associated noncompact-ness criterion of Theorem 2.1(b), and K0 is an open compact subgroup of G(Afin),we have a G(R)-isomorphism of Hilbert spaces

L2(G(Q)\G(A)/K0

) = ni=1

L2(i\G(R)

),

for discrete subgroups 1, . . . ,n of G(R). The Hecke algebra H(G(Afin),K0

)acts

by convolution on the left hand side, and hence also on the right hand side.Hecke operators are really at the heart of the theory. Their properties can be

formulated in representation theoretic terms. Any automorphic representation ofG(A) can be decomposed as a restricted tensor product

(2.2) =

v

v,

where v is an irreducible representation of the group G(Qv). Moreover, for everyvaluation v = vp outside some finite set S, the representation p = vp is unramified,in the sense that its restriction to a suitable maximal compact subgroup Kp ofG(Qp) contains the trivial representation. (See [F]. It is known that the trivialrepresentation of Kp occurs in p with multiplicity at most one.) This gives rise toa maximal compact subgroup KS =

p/S

Kp, a Hecke algebra

HS =p/S

Hp =p/S

H(G(Qp),Kp

)that is actually abelian, and an algebra homomorphism

(2.3) c(S) =p/S

c(p) : HS =p/S

Hp C.

14 JAMES ARTHUR

Indeed, if vS =p/S

vp belongs to the one-dimensional space of KS-fixed vectors for

the representation S =p/S

p, and hS =p/S

hp belongs to HS , the vector

S(hS)vS =p/S

(p(hp)vp

)equals

c(S , hS)vS =p/S

(c(p, hp)vp

).

This formula defines the homomorphism (2.3) in terms of the unramified represen-tation S . Conversely, for any homomorphism HS C, it is easy to see that thereis a unique unramified representation S of G(AS) for which the formula holds.

The decomposition (2.2) actually holds for general irreducible representations of G(A). In this case, the components can be arbitrary. However, the conditionthat be automorphic is highly rigid. It imposes deep relationships among thedifferent unramified components p, or equivalently, the different homomorphismsc(p) : Hp C. These relationships are expected to be of fundamental arithmeticsignificance. They are summarized by Langlandss principle of functoriality [Lan3],and his conjecture that relates automorphic representations to motives [Lan7].(For an elementary introduction to these conjectures, see [A28]. We shall reviewthe principle of functoriality and its relationship with unramified representationsin 26.) The general trace formula provides a means for analyzing some of therelationships.

The group G(A) can be written as a direct product of the real group G(R) withthe totally disconnected group G(Afin). We define

Cc(G(A)

)= Cc

(G(R)

) Cc

(G(Afin)

),

where Cc(G(R)

)is the usual space of smooth, compactly supported functions on

the Lie group G(R), and Cc(G(Afin)

)is the space of locally constant, compactly

supported, complex valued functions on the totally disconnected group G(Afin).The vector space Cc

(G(A)

)is an algebra under convolution, which is of course

contained in the algebra Cc(G(A)

)of continuous, compactly supported functions

on G(A).Suppose that f belongs to Cc

(G(A)

). We can choose a finite set of valuations

S satisfying the condition of Theorem 2.1(b), an open compact subgroup KS ofG(AS), and an open compact subgroup K0,S of the product

G(QS ) =

vS{v}G(Qv)

such that f is bi-invariant under the open compact subgroup K0 = K0,SKS ofG(Afin). In particular, the operator R(f) vanishes on the orthogonal complementof L2

(G(Q)\G(A)/KS

)in L2

(G(Q)\G(A)

). We leave the reader the exercise of

using (1.1) and (2.1) to identify R(f) with an integral operator with smooth kernelon a finite disjoint union of quotients of G(R).

Suppose, in particular, that G(Q)\G(A) happens to be compact. Then R(f)may be identified with an integral operator with smooth kernel on a compact man-ifold. It follows that R(f) is an operator of trace class, whose trace is given by

3. SIMPLE EXAMPLES 15

(1.2). The Selberg trace formula (1.3) is therefore valid for f , with = G(Q) andH = G(A). (See [Tam1].)

3. Simple examples

We have tried to introduce adele groups as gently as possible, using the re-lations between Hecke operators and automorphic representations as motivation.Nevertheless, for a reader unfamiliar with such matters, it might take some time tofeel comfortable with the general theory. To supplement the discussion of 2, andto acquire some sense of what one might hope to obtain in general, we shall lookat a few concrete examples.

Consider first the simplest example of all, the case that G equals the multi-plicative group GL(1). Then G(Q) = Q, while

G(A) = A ={x A : |x| = 0, |xp|p = 1 for almost all p

}is the multiplicative group of ideles for Q. If N is a positive integer with primefactorization N =

ppep(N), we write

KN ={k G(Afin) = Afin : |kp 1|p pep(N) for all p

}.

A simple exercise for a reader unfamiliar with adeles is to check directly that KNis an open compact subgroup of Afin, that any open compact subgroup K0 containsKN for some N , and that the abelian group

G(Q)\G(A)/G(R)KN = Q\A/RKNis finite. The quotient G(Q)\G(A) = Q\A is not compact. This is because themapping

x |x| =v

|xv|v, x A,

is a continuous surjective homomorphism from A to the multiplicative group (R)0

of positive real numbers, whose kernel

A1 ={x A : |x| = 1

}contains Q. The quotient Q\A1 is compact. Moreover, we can write the groupA as a canonical direct product of A1 with the group (R)0. The failure of Q\Ato be compact is therefore entirely governed by the multiplicative group (R)0 ofpositive real numbers.

An irreducible unitary representation of the abelian group GL(1,A) = A is ahomomorphism

: A U(1) ={z C : |z| = 1

}.

There is a free action

s : s(x) = (x)|x|s, s iR,of the additive group iR on the set of such . The orbits of iR are bijectiveunder the restriction mapping from A to A1 with the set of irreducible unitaryrepresentations of A1. A similar statement applies to the larger set of irreducible(not necessarily unitary) representations of A, except that one has to replace iRwith the additive group C.

Returning to the case of a general group over Q, we write AG for the largest cen-tral subgroup of G over Q that is a Q-split torus. In other words, AG is Q-isomorphic

16 JAMES ARTHUR

to a direct product GL(1)k of several copies of GL(1). The connected componentAG(R)0 of 1 in AG(R) is isomorphic to the multiplicative group

((R)0

)k, whichin turn is isomorphic to the additive group Rk. We write X(G)Q for the additivegroup of homomorphisms : g g from G to GL(1) that are defined over Q.Then X(G)Q is a free abelian group of rank k. We also form the real vector space

aG = HomZ(X(G)Q,R

)of dimension k. There is then a surjective homomorphism

HG : G(A) aG,defined by

HG(x),

= log(x), x G(A), X(G)Q.

The group G(A) is a direct product of the normal subgroup

G(A)1 ={x G(A) : HG(x) = 0

}with AG(R)0.

We also have the dual vector space aG = X(G)QZ R, and its complexificationaG,C = X(G)Q C. If is an irreducible unitary representation of G(A) and belongs to iaG, the product

(x) = (x)e(HG(x)), x G(A),is another irreducible unitary representation of G(A). The set of associated iaG-orbits is in bijective correspondence under the restriction mapping from G(A) toG(A)1 with the set of irreducible unitary representations of G(A)1. A similar as-sertion applies the larger set of irreducible (not necessary unitary) representations,except that one has to replace iaG with the complex vector space a

G,C.

In the case G = GL(n), for example, we have

AGL(n) =

z 0. . .

0 z

: z GL(1) = GL(1).

The abelian group X(GL(n)

)Q

is isomorphic to Z, with canonical generator givenby the determinant mapping from GL(n) to GL(1). The adelic group GL(n,A) isa direct product of the two groups

GL(n,A)1 ={x GL(n,A) : | det(x)| = 1

}and

AGL(n)(R)0 =

r 0. . .

0 r

: r (R)0 .

In general, G(Q) is contained in the subgroup G(A)1 of G(A). The groupAG(R)0 is therefore an immediate obstruction to G(Q)\G(A) being compact, asindeed it was in the simplest example of G = GL(1). The real question is thenwhether the quotient G(Q)\G(A)1 is compact. When the answer is affirmative, thediscussion above tells us that the trace formula (1.3) can be applied. It holds for = G(Q) and H = G(A)1, with f being the restriction to G(A)1 of a function inCc(G(A)

).

3. SIMPLE EXAMPLES 17

The simplest nonabelian example that gives compact quotient is the multiplica-tive group

G = {x A : x = 0}of a quaternion algebra over Q. By definition, A is a four dimensional divisionalgebra over Q, with center Q. It can be written in the form

A ={x = x0 + x1i + x2j + x3k : x Q

},

where the basis elements 1, i, j and k satisfy

ij = ji = k, i2 = a, j2 = b,for nonzero elements a, b Q. Conversely, for any pair a, b Q, the Q-algebradefined in this way is either a quaternion algebra or is isomorphic to the matrixalgebra M2(Q). For example, if a = b = 1, A is a quaternion algebra, sinceAQ R is the classical Hamiltonian quaternion algebra over R. On the other hand,if a = b = 1, the mapping

x x0(

1 00 1

)+ x1

(1 00 1

)+ x2

(0 11 0

)+ x3

(0 11 0

)is an isomorphism from A onto M2(Q). For any A, one defines an automorphism

x x = x0 x1i x2j x3kof A, and a multiplicative mapping

x N(x) = xx = x0 ax21 bx22 + abx23from A to Q. If N(x) = 0, x1 equals N(x)1x. It follows that x A is a unit ifand only if N(x) = 0.

The description of a quaternion algebra A in terms of rational numbers a, b Qhas the obvious attraction of being explicit. However, it is ultimately unsatisfactory.Among other things, different pairs a and b can yield the same algebra A. Thereis a more canonical characterization in terms of the completions Av = AQ Qv atvaluations v of Q. If v = v, we know that Av is isomorphic to either the matrixring M2(R) or the Hamiltonian quaternion algebra over R. A similar propertyholds for any other v. Namely, there is exactly one isomorphism class of quaternionalgebras over Qv, so there are again two possibilities for Av. Let V be the set ofvaluations v such that Av is a quaternion algebra. It is then known that V is afinite set of even order. Conversely, for any nonempty set V of even order, thereis a unique isomorphism class of quaternion algebras A over Q such that Av is aquaternion algebra for each v V and a matrix algebra M2(Qv) for each v outsideV .

We digress for a moment to note that this characterization of quaternion al-gebras is part of a larger classification of reductive algebraic groups. The generalclassification over a number field F , and its completions Fv, is a beautiful union ofclass field theory with the structure theory of reductive groups. One begins with agroup Gs over F that is split, in the sense that it has a maximal torus that splitsover F . By a basic theorem of Chevalley, the groups Gs are in bijective correspon-dence with reductive groups over an algebraic closure F of F , the classificationof which reduces largely to that of complex semisimple Lie algebras. The generalgroup G over F is obtained from Gs by twisting the action of the Galois groupGal(F/F ) by automorphisms of Gs . It is a two stage process. One first constructs

18 JAMES ARTHUR

an outer twist G of Gs that is quasisplit, in the sense that it has a Borel sub-group that is defined over F . This is the easier step. It reduces to a knowledgeof the group of outer automorphisms of Gs , something that is easy to describe interms of the general structure of reductive groups. One then constructs an innertwist G

G, where is an isomorphism such that for each Gal(F/F ), thecomposition

() = ()1

belongs to the group Int(G) of inner automorphisms of G. The role of class fieldtheory is to classify the functions (). More precisely, class field theoryallows us to characterize the equivalence classes of such functions defined by theGalois cohomology set

H1(F, Int(G)

)= H1

(Gal(F/F ), Int(G)(F )

).

It provides a classification of the finite sets of local inner twists H1(Fv, Int(Gv)

),

and a characterization of the image of the map

H1(F, Int(G)

)v

H1(F, Int(Gv)

)in terms of an explicit generalization of the parity condition for quaternion algebras.The map is injective, by the Hasse principle for the adjoint group Int(G). Its imagetherefore classifies the isomorphism classes of inner twists G of G over F .

In the special case above, the classification of quaternion algebras A is equiva-lent to that of the algebraic groups A. In this case, G = Gs = GL(2). In general,the theory is not especially well known, and goes beyond what we are assuming forthis course. However, as a structural foundation for the Langlands program, it iswell worth learning. A concise reference for a part of the theory is [Ko5, 1-2].

Let G be the multiplicative group of a quaternion algebra A over Q, as above.The restriction of the norm mapping N to G is a generator of the group X(G)Q.In particular,

G(A)1 ={x G(A) : |N(x)| = 1

}.

It is then not hard to see that the quotient G(Q)\G(A)1 is compact. (The reasonis that G has no proper parabolic subgroup over Q, a point we shall discuss inthe next section.) The Selberg trace formula (1.3) therefore holds for = G(Q),H = G(A)1, and f the restriction to G(A)1 of a function in Cc

(G(A)

). If (G)

denotes the set of conjugacy classes in G(Q), and (G) is the set of equivalenceclasses of automorphic representations of G (or more properly, restrictions to G(A)1

of automorphic representations of G(A)), we have

(3.1)

(G)aG()fG() =

(G)

aG()fG(), f Cc(G(A)

),

for the volume aG() = aH (), the multiplicity aG() = aH (), the orbital integral

fG() = fH(), and the character fG() = fH(). Jacquet and Langlands gave astriking application of this formula in 16 of their monograph [JL].

Any function in Cc(G(A)

)is a finite linear combination of products

f =v

fv, fv Cc(G(Qv)

).

Assume that f is of this form. Then fG() is a product of local orbital integralsfv,G(v), where v is the image of in the set (Gv) of conjugacy classes in G(Qv),

3. SIMPLE EXAMPLES 19

and fG() is a product of local characters fv,G(v), where v is the component of in the set (Gv) of equivalence classes of irreducible representations of G(Qv).Let V be the even set of valuations v such that G is not isomorphic to the groupG = GL(2) over Qv. If v does not belong to V , the Qv-isomorphism from G toG is determined up to inner automorphisms. There is consequently a canonicalbijection v v from (Gv) to (Gv), and a canonical bijection v v from(Gv) to (Gv). One can therefore define a function f

v Cc (Gv) for every v / V

such thatfv,G(

v) = fv,G(v)

andfv,G(

v) = fv,G(v),

for every v (Gv) and v (Gv). This suggested to Jacquet and Langlandsthe possibility of comparing (3.1) with the trace formula Selberg had obtained forthe group G = GL(2) with noncompact quotient.

If v belongs to V , G(Qv) is the multiplicative group of a quaternion algebraover Qv. In this case, there is a canonical bijection v v from (Gv) onto theset ell(Gv) of semisimple conjugacy classes in G(Qv) that are either central, ordo not have eigenvalues in Qv. Moreover, there is a global bijection from(G) onto the set of semisimple conjugacy classes (G) such that for everyv V , v belongs to ell(Gv). For each v V , Jacquet and Langlands assigned afunction fv Cc

(G(Qv)

)to fv such that

(3.2) fv,G(v) =

{fv,G(v), if v ell(Gv),0, otherwise,

for every (strongly) regular class v reg(Gv). (An element is strongly regular ifits centralizer is a maximal torus. The strongly regular orbital integrals of fv areknown to determine the value taken by fv at any invariant distribution on G(Qv).)This allowed them to attach a function

f =v

fv

in Cc(G(A)

)to the original function f . They then observed that

(3.3) fG() =

{fG(), if is the image of (G),0, otherwise,

for any class (G).It happens that Selbergs formula for the group G = GL(2) contains a number

of supplementary terms, in addition to analogues of the terms in (3.1). However,Jacquet and Langlands observed that the local vanishing conditions (3.2) force allof the supplementary terms to vanish. They then used (3.3) to deduce that theremaining terms on the geometric side equaled the corresponding terms on thegeometric side of (3.1). This left only a spectral identity

(3.4)

(G)m(,R)tr

((f)

)=

(G)

m(, Rdisc)tr((f)

),

where Rdisc is the subrepresentation of the regular representation of G(A)1 on

L2(G(Q)\G(A)1

)that decomposes discretely. By setting f = fSfS , for a fixed fi-

nite set S of valuations containing V {v}, and a fixed function fS Cc(G(QS)

),

20 JAMES ARTHUR

one can treat (3.4) as an identity of linear forms in a variable function fS belong-ing to the Hecke algebra H(GS ,KS). Jacquet and Langlands used it to establishan injective global correspondence of automorphic representations, withv = v for each v / V . They also obtained an injective local correspondencev v of irreducible representations for each v V , which is compatible withthe global correspondence, and also the local correspondence fv fv of functions.Finally, they gave a simple description of the images of both the local and globalcorrespondences of representations.

The Jacquet-Langlands correspondence is remarkable for both the power of itsassertions and the simplicity of its proof. It tells us that the arithmetic informationcarried by unramified components p of automorphic representations of G(A),whatever form it might take, is included in the information carried by automorphicrepresentations of G(A). In the case v / V , it also implies a correspondencebetween spectra of Laplacians on certain compact Riemann surfaces, and discretespectra of Laplacians on noncompact surfaces. The Jacquet-Langlands correspon-dence is a simple prototype of the higher reciprocity laws one might hope to deducefrom the trace formula. In particular, it is a clear illustration of the importance ofhaving a trace formula for noncompact quotient.

4. Noncompact quotient and parabolic subgroups

If G(Q)\G(A)1 is not compact, the two properties that allowed us to derivethe trace formula (1.3) fail. The regular representation R does not decomposediscretely, and the operators R(f) are not of trace class. The two properties areclosely related, and are responsible for the fact that the integral (1.2) generallydiverges. To see what goes wrong, consider the case that G = GL(2), and take fto be the restriction to H = G(A)1 of a nonnegative function in Cc

(G(A)

). If the

integral (1.2) were to converge, the double integralG(Q)\G(A)1

G(Q)

f(x1x)dx

would be finite. Using Fubinis theorem to justify again the manipulations of 1,we would then be able to write the double integral as

{G(Q)}vol(G(Q)\G(A)1

) G(A)1\G(A)1

f(x1x)dx.

As it happens, however, the summand corresponding to is often infinite.

Sometimes the volume of G(Q)\G(A)1 is infinite. Suppose that =(1 00 2

),

for a pair of distinct elements 1 and 2 in Q. Then

G ={(

y1 00 y2

): y1, y2 GL(1)

}= GL(1)GL(1),

so thatG(A)1 =

{(y1, y2) (A)2 : |y1||y2| = 1

},

andG(Q)\G(A)1 = (Q\A1) (Q\A) = (Q\A1)2 (R)0.

4. NONCOMPACT QUOTIENT AND PARABOLIC SUBGROUPS 21

An invariant measure on the left hand quotient therefore corresponds to a Haarmeasure on the abelian group on the right. Since this group is noncompact, thequotient has infinite volume.

Sometimes the integral over G(A)1\G(A)1 diverges. Suppose that =(

1 10 1

).

Then

G(A) ={(

z y0 z

): y A, z A

}The computation of the integral

G(A)1\G(A)1f(x1x)dx =

G(A)\G(A)

f(x1x)dx

is a good exercise in understanding relations among the Haar measures da, du anddx on A, A, and G(A), respectively. One finds that the integral equals

G(A)\P0(A)

P0(A)\G(A)

f(k1p1pk)dpdk,

where P0(A) is the subgroup of upper triangular matrices{p =

(a u0 b

): a, b A, u A

},

with left Haar measuredp = |a|1dadbdu,

and dk is a Borel measure on the compact space P0(A)\G(A). The integral thenreduces to an expression

c(f)p

(1 p1)1 = c(f)(

n=1

1n

),

where

c(f) = c0

P0(A)\G(A)

A

f

(k1

(1 u0 1

)k

)dudk,

for a positive constant c0. In particular, the integral is generally infinite.Observe that the nonconvergent terms in the case G = GL(2) both come from

conjugacy classes in GL(2,Q) that intersect the parabolic subgroup P0 of uppertriangular matrices. This suggests that rational parabolic subgroups are responsiblefor the difficulties encountered in dealing with noncompact quotient. Our suspicionis reinforced by the following characterization, discovered independently by Boreland Harish-Chandra [BH] and Mostow and Tamagawa [MT]. For a general groupG over Q, the quotient G(Q)\G(A)1 is noncompact if and only if G has a properparabolic subgroup P defined over Q.

We review some basic properties of parabolic subgroups, many of which arediscussed in the chapter [Mur] in this volume. We are assuming now that Gis a general connected reductive group over Q. A parabolic subgroup of G is analgebraic subgroup P such that P (C)\G(C) is compact. We consider only parabolicsubgroups P that are defined over Q. Any such P has a Levi decomposition P =MNP , which is a semidirect product of a reductive subgroup M of G over Qwith a normal unipotent subgroup NP of G over Q. The unipotent radical NP isuniquely determined by P , while the Levi component M is uniquely determined upto conjugation by P (Q).

22 JAMES ARTHUR

Let P0 be a fixed minimal parabolic subgroup of G over Q, with a fixed Levidecomposition P0 = M0N0. Any subgroup P of G that contains P0 is a parabolicsubgroup that is defined over Q. It is called a standard parabolic subgroup (relativeto P0). The set of standard parabolic subgroups of G is finite, and is a set ofrepresentatives of the set of all G(Q)-conjugacy classes of parabolic subgroups overQ. A standard parabolic subgroup P has a canonical Levi decomposition P =MPNP , where MP is the unique Levi component of P that contains M0. GivenP , we can form the central subgroup AP = AMP of MP , the real vector spaceaP = aMP , and the surjective homomorphism HP = HMP from MP (A) onto aP .In case P = P0, we often write A0 = AP0 , a0 = aP0 and H0 = HP0 .

In the example G = GL(n), one takes P0 to be the Borel subgroup of uppertriangular matrices. The unipotent radical N0 of P0 is the subgroup of unipotentupper triangular matrices. For the Levi component M0, one takes the subgroup ofdiagonal matrices. There is then a bijection

P (n1, . . . , np)

between standard parabolic subgroups P of G = GL(n) and partitions (n1, . . . , np)of n. The group P is the subgroup of block upper triangular matrices associatedto (n1, . . . , np). The unipotent radical of P is the corresponding subgroup

NP =

In1 | . . .

0 |Inp

of block unipotent matrices, the canonical Levi component is the subgroup

MP =

m =m1| 0. . .

0 |mp

: mi GL(ni)

of block diagonal matrices, while

AP =

a =a1In1 | 0. . .

0 |apInp

: ai GL(1) .

Naturally, Ik stands here for the identity matrix of rank k. The free abelian groupX(MP )Q attached to MP has a canonical basis of rational characters

i : m det(mi), m MP , 1 i p.

We are free to use the basis 1n11, . . . ,1

npp of the vector space aP , and the corre-

sponding dual basis of aP , to identify both aP and aP with Rp. With this interpre-

tation, the mapping HP takes the form

HP (m) =(

1n1

log | det m1|, . . . ,1np

log | det mp|), m MP (A).

It follows that

HP (a) =(log |a1|, . . . , log |ap|

), a AP (A).

4. NONCOMPACT QUOTIENT AND PARABOLIC SUBGROUPS 23

For general G, we have a variant of the regular representation R for anystandard parabolic subgroup P . It is the regular representation RP of G(A) onL2(NP (A)MP (Q)\G(A)

), defined by(

RP (y))(x) = (xy), L2

(NP (A)MP (Q)\G(A)

), x, y G(A).

Using the language of induced representations, we can write

RP = IndG(A)NP (A)MP (Q)

(1NP (A)MP (Q)) = IndG(A)P (A)(1NP (A) RMP ),

where IndHK() denotes a representation of H induced from a subgroup K, and 1Kdenotes the trivial one dimensional representation of K. We can of course integrateRP against any function f Cc

(G(A)

). This gives an operator RP (f) on the

Hilbert space L2(NP (A)MP (Q)\G(A)

). Arguing as in the special case R = RG of

1, we find that RP (f) is an integral operator with kernel

(4.1) KP (x, y) =

NP (A)

MP (Q)

f(x1ny)dn, x, y NP (A)MP (Q)\G(A).

We have seen that the diagonal value K(x, x) = KG(x, x) of the original kernelneed not be integrable over x G(Q)\G(A)1. We have also suggested that parabolicsubgroups are somehow responsible for this failure. It makes sense to try to modifyK(x, x) by adding correction terms indexed by proper parabolic subgroups P . Thecorrection terms ought to be supported on some small neighbourhood of infinity, sothat they do not affect the values taken by K(x, x) on some large compact subsetof G(Q)\G(A)1. The diagonal value KP (x, x) of the kernel of RP (f) provides anatural function for any P . However, KP (x, x) is invariant under left translationof x by the group NP (A)MP (Q), rather than G(Q). One could try to rectify thisdefect by summing KP (x, x) over elements in P (Q)\G(Q). However, this sumdoes not generally converge. Even if it did, the resulting function on G(Q)\G(A)1would not be supported on a small neighbourhood of infinity. The way aroundthis difficulty will be to multiply KP (x, x) by a certain characteristic function onNP (A)MP (Q)\G(A) that is supported on a small neighbourhood of infinity, andwhich depends on a choice of maximal compact subgroup K of G(A).

In case G = GL(n), the product

K = O(n,R)p

GL(n,Zp)

is a maximal compact subgroup of G(A). According to the Gramm-Schmidt or-thogonalization lemma of linear algebra, we can write

GL(n,R) = P0(R)O(n,R).

A variant of this process, applied to the height function

vp = max{|vi|p : 1 i n}, v Qnp ,

on Qnp instead of the standard inner product on Rn, gives a decomposition

GL(n,Qp) = P0(Qp)GL(n,Zp),

for any p. It follows that GL(n,A) equals P0(A)K.

24 JAMES ARTHUR

These properties carry over to our general group G. We choose a suitablemaximal compact subgroup

K =v

Kv, Kv G(Qv),

of G(A), with G(A) = P0(A)K [Ti, (3.3.2), (3.9], [A5, p. 9]. We fix K, andconsider a standard parabolic subgroup P of G. Since P contains P0, we obtain adecomposition

G(A) = P (A)K = NP (A)MP (A)K = NP (A)MP (A)1AP (R)0K.

We then define a continuous mapping

HP : G(A) aPby setting

HP (nmk) = HMP (m), n NP (A), m MP (A), k K.We shall multiply the kernel KP (x, x) by the preimage under HP of the character-istic function of a certain cone in aP .

5. Roots and weights

We have fixed a minimal parabolic subgroup P0 of G, and a maximal compactsubgroup K of G(A). We want to use these objects to modify the kernel functionK(x, x) so that it becomes integrable. To prepare for the construction, as well asfor future geometric arguments, we review some properties of roots and weights.

The restriction homomorphism X(G)Q X(AG)Q is injective, and has finitecokernel. If G = GL(n), for example, the homomorphism corresponds to the injec-tion z nz of Z into itself. We therefore obtain a canonical linear isomorphism(5.1) aP = X(MP )Q R

X(AP )Q R.Now suppose that P1 and P2 are two standard parabolic subgroups, with P1

P2. There are then Q-rational embeddings

AP2 AP1 MP1 MP2 .The restriction homomorphism X(MP2)Q X(MP1)Q is injective. It providesa linear injection aP2 a

P1

and a dual linear surjection aP1 aP2 . We writeaP2P1 aP1 for the kernel of the latter mapping. The restriction homomorphismX(AP1)Q X(AP2)Q is surjective, and extends to a surjective mapping fromX(AP1)Q R to X(AP2)Q R. It thus provides a linear surjection aP1 a

P2

,and a dual linear injection aP2 aP1 . Taken together, the four linear mappingsyield split exact sequences

0 aP2 aP1 a

P1/a

P2 0

and0 aP2P1 aP1 aP2 0

of real vector spaces. We may therefore write

aP1 = aP2 aP2P1and

aP1 = aP2 (a

P2P1

).

5. ROOTS AND WEIGHTS 25

For any P , we write P for the set of roots of (P,AP ). We also write nP forthe Lie algebra of NP . Then P is a finite subset of nonzero elements in X(AP )Qthat parametrizes the decomposition

nP =

P

n

of nP into eigenspaces under the adjoint action

Ad : AP GL(nP )

of AP . By definition,

n ={X nP : Ad(a)X = aX, a AP

},

for any P . We identify P with a subset of aP under the canonical mappings

P X(AP )Q X(AP )Q R aP .

If H belongs to the subspace aG of aP , (H) = 0 for each P , so P iscontained in the subspace (aGP )

of aP . As is customary, we define a vector

P =12

P

(dim n)

in (aGP ). We leave the reader to check that left and right Haar measures on the

group P (A) are related by

dp = e2(HP (p))drp, p P (A).

In particular, the group P (A) is not unimodular, if P = G.We write 0 = P0 . The pair

(V,R) =((aGP0)

,0 (0))

is a root system [Ser2], for which 0 is a system of positive roots. We writeW0 = WG0 for the Weyl group of (V,R). It is the finite group generated by reflectionsabout elements in 0, and acts on the vector spaces V = (aGP0)

, a0 = aP0

, anda0 = aP0 . We also write 0 0 for the set of simple roots attached to 0. Then0 is a basis of the real vector space (aG0 )

= (aGP0). Any element 0 can be

written uniquely

=

0

n,

for nonnegative integers n. The corresponding set

0 = { : 0}

of simple coroots is a basis of the vector space aG0 = aGP0

. We write

0 = { : 0}

for the set of simple weights, and

0 = { : 0}

for the set of simple co-weights. In other words, 0 is the basis of (aG0 ) dual to

0 , and 0 is the basis of aG0 dual to 0.

26 JAMES ARTHUR

Standard parabolic subgroups are parametrized by subsets of 0. More pre-cisely, there is an order reversing bijection P P0 between standard parabolicsubgroups P of G and subsets P0 of 0, such that

aP ={H a0 : (H) = 0, P0

}.

For any P , P0 is a basis of the space aPP0

= aP0 . Let P be the set of linear formson aP obtained by restriction of elements in the complement 0P0 of P0 in 0.Then P is bijective with 0 P0 , and any root in P can be written uniquelyas a nonnegative integral linear combination of elements in P . The set P is abasis of (aGP )

. We obtain a second basis of (aGP ) by taking the subset

P = { : 0 P0 }of 0. We shall write

P = { : P }for the basis of aGP dual to P , and

P = { : P }for the basis of aGP dual to P . We should point out that this notation is notstandard if P = P0. For in this case, a general element P is not part of aroot system (as defined in [Ser2]), so that is not a coroot. Rather, if is therestriction to aP of the simple root 0 P0 , is the projection onto aP ofthe coroot .

We have constructed two bases P and P of (aGP ), and corresponding dual

bases P and P of a

GP , for any P . More generally, suppose that P1 P2 are two

standard parabolic subgroups. Then we can form two bases P2P1 and P2P1

of (aP2P1),

and corresponding dual bases (P2P1) and (P2P1)

of aP2P1 . The construction proceedsin the obvious way from the bases we have already defined. For example, P2P1 isthe set of linear forms on the subspace aP2P1 of aP1 obtained by restricting elementsin P20 P10 , while P2P1 is the set of linear forms on a

P2P1

obtained by restrictingelements in P1 P2 . We note that P1MP2 is a standard parabolic subgroup ofthe reductive group MP2 , relative to the fixed minimal parabolic subgroup P0MP2 .It follows from the definitions that

aP1MP2 = aP1 , aMP2P1MP2

= aP2P1 , P1MP2 = P2P1,

andP1MP2 =

P2P1.

Consider again the example of G = GL(n). Its Lie algebra is the space Mn of(n n)-matrices, with the Lie bracket

[X,Y ] = XY Y X,and the adjoint action

Ad(g) : X gXg1, g G, X Mn,of G. The group

A0 =

a =a1 0. . .

0 an

: ai GL(1)

5. ROOTS AND WEIGHTS 27

acts by conjugation on the Lie algebra

n0 = nP0 =

0

. . . . . ....

. . . 0 0

of NP0 , and0 = {ij : a aia1j , i < j}.

As linear functionals on the vector space

a0 =

u :u1 0. . .

0 un

: ui R ,

the roots 0 take the form

ij(u) = ui uj , i < j.The decomposition of a general root in terms of the subset

0 = {i = i,i+1 : 1 i n 1},of simple roots is given by

ij = i + + j1, i < j.The set of coroots equals

0 ={ij = ei ej = (

j 0, . . . , 0, 1

i

, 0, . . . , 0,1, 0, . . . , 0) : i < j},

where we have identified a0 with the vector space Rn, equipped with the standardbasis e1, . . . , en. The simple coroots form the basis

0 = {i = ei ei+1 : 1 i n 1}of the subspace

aG0 = {u Rn :

ui = 0}.The simple weights give the dual basis

0 = {i : 1 i n 1},where

i(u) =n in

(u1 + + ui)( in

)(ui+1 + + un).

The Weyl group W0 of the root system for GL(n) is the symmetric group Sn, actingby permutation of the coordinates of vectors in the space a0 = Rn. The dot producton Rn give a W -invariant inner product , on both a0 and a0. It is obvious that

i, j 0, i = j.We leave to the reader the exercise of showing that

i, j 0, 1 i, j n 1.

28 JAMES ARTHUR

Suppose that P GL(n) corresponds to the partition (n1, . . . , np) of n. Thegeneral embedding aP a0 we have defined corresponds to the embedding

t (t1, . . . , t1

n1

, t2, . . . , t2 n2

, . . . , tp, . . . , tp np

), t Rp,

of Rp into Rn. It follows that

P0 = {i : i = n1 + + nk, 1 k p 1}.Since P is the set of restrictions to aP a0 of elements in the set

0 P0 = {n1 , n1+n2 , . . .},we see that

P = {i : t ti ti+1, 1 i p 1, t Rp}.The example of G = GL(n) provides algebraic intuition. It is useful for readers

less familiar with general algebraic groups. However, the truncation of the kernelalso requires geometric intuition. For this, the example of G = SL(3) is oftensufficient.

The root system for SL(3) is the same as for GL(3). In other words, we canidentify a0 with the two dimensional subspace

{u R3 :

ui = 0}

of R3, in which case

0 = {1, 2} 0 = {1, 2, 1 + 2},in the notation above. We can also identify a0 isometrically with the two dimensionEuclidean plane. The singular (one-dimensional) hyperplanes, the coroots 0 , andthe simple coweights (0) are then illustrated in the familiar Figures 5.1 and 5.2.

1

2

1 + 2

aP1

aP2

Figure 5.1. The two simple coroots 1 and 2 are orthogonal to

the respective subspaces aP2 and aP1 of a0. Their inner product isnegative, and they span an obtuse angled cone.

There are four standard parabolic subgroups P0, P1, P2, and G, with P1 and P2being the maximal parabolic subgroups such that P10 = {2} and P20 = {1}.

6. STATEMENT AND DISCUSSION OF A THEOREM 29

1

2

Figure 5.2. The two simple coweights 1 and 2 lie in the respec-

tive subspaces aP1 and aP2 . Their inner product is positive, and theyspan an acute angled cone.

6. Statement and discussion of a theorem

Returning to the general case, we can now describe how to modify the functionK(x, x) on G(Q)\G(A). For a given standard parabolic subgroup P , we write Pfor the characteristic function of the subset

a+P = {t aP : (t) > 0, P }of aP . In the case G = SL(3), this subset is the open cone generated by 1 and2 in Figure 5.2 above. We also write P for the characteristic function of thesubset

{t aP : (t) > 0, P }of aP . In case G = SL(3), this subset is the open cone generated by 1 and

2 in

Figure 5.1.The truncation of K(x, x) depends on a parameter T in the cone a+0 = a

+P0

thatis suitably regular, in the sense that (T ) is large for each root 0. For anygiven T , we define(6.1)kT (x) = kT (x, f) =

P

(1)dim(AP /AG)

P (Q)\G(Q)KP (x, x)P

(HP (x) T

).

This is the modified kernel, on which the general trace formula is based. A fewremarks might help to put it into perspective.

One has to show that for any x, the sum over in (6.1) may be taken over afinite set. In the case G = SL(2), the reader can verify the property as an exercise inreduction theory for modular forms. In general, it is a straightforward consequence[A3, Lemma 5.1] of the Bruhat decomposition for G and the construction by Boreland Harish-Chandra of an approximate fundamental domain for G(Q)\G(A). (Weshall recall both of these results later.) Thus, kT (x) is given by a double sum over(P, ) in a finite set. It is a well defined function of x G(Q)\G(A).

Observe that the term in (6.1) corresponding to P = G is just K(x, x). Incase G(Q)\G(A)1 is compact, there are no proper parabolic subgroups P (over

30 JAMES ARTHUR

Q). Therefore kT (x) equals K(x, x) in this case, and the truncation operation istrivial. In general, the terms with P = G represent functions on G(Q)\G(A)1 thatare supported on some neighbourhood of infinity. Otherwise said, kT (x) equalsK(x, x) for x in some large compact subset of G(Q)\G(A)1 that depends on T .

Recall that G(A) is a direct product of G(A)1 with AG(R)0. Observe also thatkT (x) is invariant under translation of x by AG(R)0. It therefore suffices to studykT (x) as a function of x in G(Q)\G(A)1.

Theorem 6.1. The integral

(6.2) JT (f) =

G(Q)\G(A)1kT (x, f)dx

converges absolutely.

Theorem 6.1 does not in itself provide a trace formula. It is really just a firststep. We are giving it a central place in our discussion for two reasons. The state-ment of the theorem serves as a reference point for outlining the general strategy.In addition, the techniques required to prove it will be an essential part of manyother arguments.

Let us pause for a moment to outline the general steps that will take us tothe end of Part I. We shall describe informally what needs to be done in order toconvert Theorem 6.1 into some semblance of a trace formula.Step 1. Find spectral expansions for the functions K(x, y) and kT (x) that areparallel to the geometric expansions (1.1) and (6.1).

This step is based on Langlandss theory of Eisenstein series. We shall describeit in the next section.Step 2. Prove Theorem 6.1.

We shall sketch the argument in 8.Step 3. Show that the function

T JT (f),defined a priori for points T a+0 that are highly regular, extends to a polynomialin T a0.

This step allows us to define JT (f) for any T a0. It turns out that there is acanonical point T0 a0, depending on the choice of K, such that the distributionJ(f) = JT0(f) is independent of the choice of P0 (though still dependent of thechoice of K). For example, if G = GL(n) and K is the standard maximal compactsubgroup of GL(n,A), T0 = 0. We shall discuss these matters in 9, making fulluse of Theorem 6.1.Step 4. Convert the expansion (6.1) of kT (x) in terms of rational conjugacy classesinto a geometric expansion of J(f) = JT0(f).

We shall give a provisional solution to this problem in 10, as a direct corollaryof the proof of Theorem 6.1.Step 5. Convert the expansion of kT (x) in 7 in terms of automorphic represen-tations into a spectral expansion of J(f) = JT0(f).

This problem turns out to be somewhat harder than the last one. We shallgive a provisional solution in 14, as an application of a truncation operator onfunctions on G(Q)\G(A)1.

7. EISENSTEIN SERIES 31

We shall call the provisional solutions we obtain for the problems of Steps 4and 5 the coarse geometric expa