arthur book endo

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The Endoscopic Classification of

Representations: Orthogonal and

Symplectic Groups

James Arthur

Department of Mathematics, University of Toronto, 40 St.

George Street, Toronto, Ontario M5S 2E4

E-mail address: [email protected]


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2000 Mathematics Subject Classification. Primary

Supported in part by NSERC Discovery Grant A3483.

Abstract.


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Contents

Preface vii

Chapter 1. Parameters 11.1. The automorphic Langlands group 11.2. Self-dual, finite dimensional representations 61.3. Representations ofGL

pN

q15

1.4. A substitute for global parameters 261.5. Statement of three main theorems 40

Chapter 2. Local Transfer 512.1. Langlands-Shelstad-Kottwitz transfer 512.2. Characterization of the local classification 622.3. Normalized intertwining operators 792.4. Statement of the local intertwining relation 962.5. Relations with Whittaker models 108

Chapter 3. Global Stabilization 1213.1. The discrete part of the trace formula 121

3.2. Stabilization 1303.3. Contribution of a parameter 1373.4. A preliminary comparison 1453.5. On the vanishing of coefficients 152

Chapter 4. The Standard Model 1654.1. Statement of the stable multiplicity formula 1654.2. On the global intertwining relation 1714.3. Spectral terms 1844.4. Endoscopic terms 1914.5. The comparison 1994.6. The two sign lemmas 212

4.7. On the global theorems 2264.8. Remarks on general groups 237

Chapter 5. A Study of Critical Cases 2515.1. On the case of square integrable 2515.2. On the case of elliptic 2625.3. A supplementary parameter ` 2705.4. Generic parameters with local constraints 285

v


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vi CONTENTS

Chapter 6. The Local Classification 2996.1. Local parameters 299

6.2. Construction of global representations 9 3076.3. Construction of global parameters 9 3196.4. The local intertwining relation for 3306.5. Orthogonality relations for 3436.6. Local packets for composite 3536.7. Local packets for simple 3656.8. Resolution 374

Chapter 7. Local Nontempered Representations 3857.1. Local parameters and duality 3857.2. Construction of global parameters 9 3947.3. The local intertwining relation for 401

7.4. Local packets for composite and simple 4137.5. Some remarks on characters 427

Chapter 8. The Global Classification 4418.1. On the global theorems 4418.2. Proof by contradiction 4518.3. Reflections on the results 4678.4. Refinements for even orthogonal groups 4828.5. An approximation of the Langlands group 500

Chapter 9. Inner Forms 5119.1. Inner twists and centralizers 5119.2. Statement of theorems for inner forms 511

9.3. Local correspondence for inner forms 5119.4. Nontempered packets for inner forms 5119.5. Completion of the proof 511

Bibliography 513


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Preface

This preface contains a brief overview of the volume. We start with arough summary of the main theorems. We then give short descriptions ofthe contents of the various chapters. At the end of the preface, we will adda couple of remarks on the overall structure of the proof, notably our useof induction. The preface can serve as an introduction. The beginning of

the actual text, in the form of the first two or three sections of Chapter 1,represents a different sort of introduction. It will be our attempt to motivatewhat follows from a few basic principles.

Automorphic representations for GLpNq have been important objects ofstudy for many years. We recall that GLpNq, the general linear group ofinvertible pN Nq-matrices, assigns a group GLpN, Rq to any commutativering R with identity. For example, R could be a number field F, or thering A AF of adeles over F. Automorphic representations of GLpNq arethe irreducible representations ofGLpN, Aq that occur in the decompositionof its regular representation on L2

`GLpN, FqzGLpN,Aq. This informal

definition is made precise in [L6], and carries over to any connected reductivegroup G over F.

The primary aim of the volume is to classify the automorphic represen-tations of special orthogonal and symplectic groups G in terms of those ofGLpNq. Our main tool will be the stable trace formula for G, which until re-cently was conditional on the fundamental lemma. The fundamental lemmahas now been established in complete generality, and in all of its variousforms. In particular, the stabilization of the trace formula is now knownfor any connected group. However, we will also require the stabilization oftwisted trace formulas for GLpNq and SOp2nq. Since these have yet to beestablished, our results will still be conditional.

A secondary purpose will be to lay foundations for the endoscopic studyof more general groups G. It is reasonable to believe that the methods weintroduce here extend to groups that Ramakrishnan has called quasiclassi-

cal. These would comprise the largest class of groups whose representationscould ultimately be tied to those of general linear groups. Our third goalis expository. In adopting a style that is sometimes more discursive thanstrictly necessary, we have tried to place at least some of the techniquesinto perspective. We hope that there will be parts of the volume that areaccessible to readers who are not experts in the subject.

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viii PREFACE

Automorphic representations are interesting for many reasons, but amongthe most fundamental is the arithmetic data they carry. Recall that

GLpN, Aq v

GLpN, Fvq

is a restricted direct product, taken over (equivalence classes of) valuationsv of F. An automorphic representation of GLpNq is a restricted directproduct

v

v,

where v is an irreducible representation of GLpN, Fvq that is unramifiedfor almost all v. We recall that v is unramified ifv is nonarchimedean, andv contains the trivial representation of the hyperspecial maximal compact

subgroup GLpN, ovq of integral points in GLpN, Fvq. The representation isthen parametrized by a semisimple conjugacy class

cvpq cpvqin the complex dual group

GLpNq^ GLpN,Cqof GLpNq. (See [Bo, (6.4), (6.5)] for the precise assertion, as it appliesto a general connected reductive group G.) It is the relations among thesemisimple conjugacy classes cvpq that will contain the fundamental arith-metic information.

There are three basic theorems for the group GLp

Nq

that together giveus a pretty clear understanding of its representations. The first is local,while others, which actually predate the first, are global.

The first theorem is the local Langlands correspondence for GLpNq. Itwas established for archimedean fields by Langlands, and more recently for p-adic (which is to say nonarchimedean) fields by Harris, Taylor and Henniart.It classifies the irreducible representations of GLpN, Fvq at all places v by(equivalence classes of) semisimple, N-dimensional representations of thelocal Langlands group

LFv #

WFv , v archimedean,

WFv SUp2q, otherwise.In particular, an unramified representation of GLpN, Fvq corresponds to anN-dimensional representation ofLFv that is trivial on the product of SUp2qwith the inertia subgroup IFv of the local Weil group WFv . It thereforecorresponds to a semisimple representation of the cyclic quotient

LFv{IFv SUp2q WFv{IFv Z,and hence a semisimple conjugacy class in GLpN,Cq, as above.


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PREFACE ix

The first of the global theorems is due to Jacquet and Shalika. If is anysmooth representation of GLpN, Aq, one can form the family of semisimpleconjugacy classes

cpq limS

cvpq cpvq : v R S

(in GLpN, Cq, defined up to a finite set of valuations S. In other words,cpq is an equivalence class of families, two such families being equivalent ifthey are equal for almost all v. The theorem of Jacquet and Shalika assertsthat if an automorphic representation of GLpNq is restricted slightly tobe isobaric [L7, 2], it is uniquely determined by cpq. This theorem canbe regarded as a generalization of the theorem of strong multiplicity one forcuspidal automorphic representations of GLpNq.

The other global theorem for GLpNq is due to Moeglin and Waldspurger.It characterizes the automorphic (relatively) discrete spectrum of

GLp

Nq interms of the set of cuspidal automorphic representations. Since Langlands

general theory of Eisenstein series characterizes the full automorphic spec-trum of any group G in terms of discrete spectra, this theorem characterizesthe automorphic spectrum for GLpNq in terms of cuspidal automorphic rep-resentations. Combined with the first global theorem, it classifies the fullautomorphic spectrum of GLpNq explicitly in terms of families cpq, con-structed from cuspidal automorphic representations of general linear groups.

Our goal is to generalize these three theorems. As we shall see, however,there is very little that comes easily. It has been known for many yearsthat the representations of groups other than GLpNq have more structure.In particular, they should separate naturally into L-packets, composed of

representations with the same L-functions and -factors. This was demon-strated for the group

G SLp2q Spp2qby Labesse and Langlands, in a paper [LL] that became a model for Lang-landss conjectural theory of endoscopy [L8], [L10].

The simplest and most elegant way to formulate the theory of endoscopyis in terms of the global Langlands group LF. This is a hypothetical globalanalogue of the explicit local Langlands groups LFv defined above. It isthought to be a locally compact extension

1 KF LF WF 1ofWF by a compact connected group KF. (See [L7, 2], [K1, 9].) However,its existence is very deep, and could well turn out to be the final theoremin the subject to be proved! One of our first tasks, which we address in1.4, will be to introduce makeshift objects to be used in place of LF. Forsimplicity, however, let us describe our results here in terms of LF.

Our main results apply to the case that G is a quasisplit special orthog-onal or symplectic group. They are stated as three theorems in 1.5. Theproof of these theorems will then take up much of the rest of the volume.


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x PREFACE

Theorem 1.5.1 is the main local result. It contains a local Langlandsparametrization of the irreducible representations of GpFvq, for any p-adicvaluation v of F, as a disjoint union of finite L-packets v . These areindexed by local Langlands parameters, namely L-homomorphisms

v : LFv LGvfrom LFv to the local L-group

LGv pG GalpFv{Fvq of G. The theoremincludes a way to index the representations in an L-packet v by linearcharacters on a finite abelian group Sv attached to v. Since similar resultsfor archimedean valuations v are already known from the work of Shelstad,we obtain a classification of the representations of each local group GpFvq.

Theorem 1.5.1 also contains a somewhat less precise description of therepresentations of GpFvq that are local components of automorphic repre-sentations. These fall naturally into rather different packets v , indexed

according to the conjectures of [A8] by L-homomorphisms

(1) v : LFv SUp2q LGv,with bounded image. The theorem includes the assertion, also conjecturedin [A8], that the representations in these packets are all unitary.

Theorem 1.5.2 is the main global result. As a first approximation, itgives a rough decomposition

(2) Rdisc

Rdisc,

of the representation Rdisc of GpAq on the automorphic discrete spectrumL

2

disc`GpFqzGpAq. The indices can be thought of as L-homomorphisms(3) : LF SUp2q LGof bounded image that do not factor through any proper parabolic subgroupof the global L-group LG pG GalpF{Fq. They have localizations (1) de-fined by conjugacy classes of embeddings LFv LF, or rather the makeshiftanalogues of such embeddings that we formulate in 1.4. The localizationsv of are unramified at almost v, and consequently lead to a family

cpq limS

cvpq cpvq : v R S

(of equivalence classes of semisimple elements in LG. The rough decompo-sition (2) is of interest as it stands. It implies that G has no embeddedeigenvalues, in the sense of unramified Hecke operators. In other words, thefamily cpq attached to any global parameter in the decomposition of theautomorphic discrete spectrum is distinct from any family obtained fromthe continuous spectrum. This follows from the nature of the parameters in (3), and the application of the theorem of Jacquet-Shalika to the naturalimage of cpq in the appropriate complex general linear group.


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xii PREFACE

acts by outer automorphism on

pG SOp2n,Cq, if G equals SOp2nq. This

is the understanding on which the decomposition (2) holds.

In the text, we shall write rpGvq for the set of equivalence classes oflocal parameters (1). The packet rv attached to any v P rpGvq will thenbe composed of rOutNpGvq-orbits of equivalence classes of irreducible rep-resentations (with rOutNpGq being trivial in case G equals SOp2n ` 1q orSpp2nq). We will write rpGq for the set of equivalence classes of generalglobal parameters , and r2pGq for the subset of classes with the supple-mentary condition of (3). The packet of any P rpGq will then be restricteddirect product

r

v

rv

of local packets. These are the objects that correspond to (isobaric) au-tomorphic representations of GLpNq. In particular, the global packets r,rather than the individual (orbits of) representations in r, are the objectsthat retain the property of strong multiplicity one from GLpNq. Similarly,the global packets r attached to parameters P r2pGq retain the quali-tative properties of automorphic discrete spectrum of GLpNq. They comewith a sort of Jordan decomposition, in which the semisimple packets cor-respond to the generic global parameters , and contain the automorphicrepresentations that are expected to satisfy the G-analogue of Ramanujansconjecture. In view of these comments, we see that Theorem 1.5.2 can beregarded as a simultaneous analogue for G of both of the global theorems

forGL

pN

q.Theorem 1.5.3 is a global supplement to Theorem 1.5.2. Its first assertionapplies to global parameters P rpGq that are both generic and simple, inthe sense that they correspond to cuspidal automorphic representations of

GLpNq. Theorem 1.5.3(a) asserts that the dual group pG is orthogonal (resp.symplectic) if and only if the symmetric square L-function Lps, , S2q (resp.the skew-symmetric square L-function Lps, , 2q) has a pole at s 1.Theorem 1.5.3(b) asserts that the Rankin-Selberg -factor

`12

, 1 2

equals 1 for any pair of generic simple parameters i P rpGiq such that pG1and pG2 are either both orthogonal or both symplectic. These two assertionsare automorphic analogues of well known properties of Artin L-functionsand -factors. They are interesting in their own right. But they are also anessential part of our induction argument. We will need them in Chapter 4 tointerpret the terms in the trace formula attached to compound parameters P rpGq.

This completes our summary of the main theorems. The first two sec-tions of Chapter 1 contain further motivation, for the global Langlands groupLF in 1.1, and the relations between representations of G and GLpNq in1.2. In 1.3, we will recall the three basic theorems for GLpNq. Section 1.4


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PREFACE xiii

is given over to our makeshift substitutes for global Langlands parameters,as we have said, while 1.5 contains the formal statements of the theorems.

As might be expected, the three theorems will have to be establishedtogether. The unified proof will take us down a long road, which starts inChapter 2, and crosses many different landscapes before coming to an endfinally in 8.2. The argument is ultimately founded on harmonic analysis,which is represented locally by orbital integrals and characters, and glob-ally by the trace formula. This of course is at the heart of the theory ofendoscopy. We refer the reader to the introductory remarks of individualsections, where we have tried to offer guidance and motivation. We shall becontent here with a minimal outline of the main stages.

Chapter 2 is devoted to local endoscopy. It contains a more preciseformulation (Theorem 2.2.1) of the local Theorem 1.5.1. This provides for a

canonical construction of the local packets rv in terms of twisted characterson GLpNq. Chapter 2 also includes the statement of Theorem 2.4.1, whichwe will call the local intertwining relation. This is closely related to Theorem1.5.1 and its refinement Theorem 2.2.1, and from a technical standpoint, canbe regarded as our primary local result. It includes a delicate constructionof signs, which will be critical for the interpretation of terms in the traceformula.

Chapter 3 is devoted to global endoscopy. We will recall the discretepart of the trace formula in 3.1, and its stabilization in 3.2. We are speak-ing here of those spectral terms that are linear combinations of automorphiccharacters, and to which all of the other terms are ultimately dedicated.They are the only terms in the trace formula that will appear explicitlyin this volume. In 3.5, we shall establish criteria for the vanishing of co-

efficients in certain identities (Proposition 3.5.1, Corollary 3.5.3). We willuse these criteria many times throughout the volume in drawing conclusionsfrom the comparison of discrete spectral terms.

In general, we will have to treat three separate cases of endoscopy. Theyare represented respectively by pairs pG, G1q, where G is one of the groupsto which Theorems 1.5.1, 1.5.2 and 1.5.3 apply and G1 is a correspondingendoscopic datum, pairs

` rGpNq, G in which rGpNq is the twisted generallinear group rG0pNq GLpNq and G is a corresponding twisted endoscopicdatum, and pairs p rG, rG1q in which rG is a twisted even orthogonal groupr

G0 SOp2nq and

rG1 is again a corresponding twisted endoscopic datum.

The first two cases will be our main concern. However, the third case

p rG, rG1

qis also a necessary part of the story. Among other things, it is forced on usby the need to specify the signs in the local intertwining relation. For themost part, we will not try to treat the three cases uniformly as cases of thegeneral theory of endoscopy. This might have been difficult, given that wehave to deduce many local and global results along the way. At any rate, theseparate treatment of the three cases gives our exposition a more concreteflavour, if at the expense of some possible sacrifice of efficiency.


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xiv PREFACE

In Chapter 4, we shall study the comparison of trace formulas. Specif-ically, we will compare the contribution (4.1.1) of a parameter to the

discrete part of the trace formula with the contribution (4.1.2) of to thecorresponding endoscopic decomposition. We begin with the statement ofTheorem 4.1.2, which we will call the stable multiplicity formula. This isclosely related to Theorem 1.5.2, and from a technical standpoint again, isour primary global result. Together with the global intertwining relation(Corollary 4.2.1), which we state as a global corollary of Theorem 2.4.1, itgoverns how individual terms in trace formulas are related. Chapter 4 rep-resents a standard model, in the sense that if we grant the analogues of thetwo primary theorems for general groups, it explains how the terms on theright hand sides of (4.1.1) and (4.1.2) match. This is discussed heuristicallyin Sections 4.7 and 4.8. However, the purpose of this volume is to deriveTheorems 2.4.1 and 4.1.2 for our groups G from the standard model, and

whatever else we can bring to bear on the problem. This is the perspectiveof Sections 4.3 and 4.4. In 4.5, we combine the analysis of these sectionswith a general induction hypothesis to deduce the stable multiplicity for-mula and the global intertwining relation for many . Section 4.6 containsthe proof of two critical sign lemmas that are essential ingredients of theparallel Sections 4.3 and 4.4.

Chapter 5 is the center of the volume. It is a bridge between the globaldiscussion of Chapters 3 and 4 and the local discussion of Chapters 6 and7. It also represents a transition from the general comparisons of Chapter 4to the study of the remaining parameters needed to complete the inductionhypotheses. These exceptional cases are the crux of the matter. In 5.2 and5.3, we shall extract several identities from the standard model, in which

we display the possible failure of Theorems 2.4.1 and 4.1.2 as correctionterms. Section 5.3 applies to the critical case of a parameter P r2pGq,and calls for the introduction of a supplementary parameter `. In 5.4,we shall resolve the global problems for families of parameters that areassumed to have certain rather technical local properties.

Chapter 6 applies to generic local parameters. It contains a proof ofthe local Langlands classification for our groups G (modified by the outerautomorphism in the case G SOp2nq). We will first have to embed a givenlocal parameter into a family of global parameters with the local constraintsof5.4. This will be the object of Sections 6.2 and 6.3, which rest ultimatelyon the simple form of the invariant trace formula for G. We will then haveto extract the required local information from the global results obtained in5.4. In 6.4, we will deduce the generic local intertwining relation from itsglobal counterpart in 5.4. Then in 6.5, we will stabilize the orthogonalityrelations that are known to hold for elliptic tempered characters. This willallow us to quantify the contributions from the remaining elliptic temperedcharacters, the ones attached to square integrable representations. We willuse the information so obtained in 6.6 and 6.7. In these sections, we shallestablish Theorems 2.2.1 and 1.5.1 for the remaining square integrable


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Langlands parameters P

r2pGq. Finally, in 6.8, we shall resolve the

various hypotheses taken on at the beginning of Chapter 6.

Chapter 7 applies to nongeneric local parameters. It contains the proofof the local theorems in general. In 7.2, we shall use the construction of6.2 to embed a given nongeneric local parameter into a family of globalparameters, but with local constraints that differ slightly from those of5.4.We will then deduce special cases of the local theorems that apply to theplaces v with local constraints. These will follow from the local theorems forgeneric parameters, established in Chapter 6, and the duality operator ofAubert and Schneider-Stuhler, which we review in 7.1. We will then exploitour control over the places v to derive the local theorems at the localization 9u that represents the original given parameter.

We will finish the proof of the global theorems in the first two sectionsof Chapter 8. Armed with the local theorems, and the resulting refinements

of the lemmas from Chapter 5, we will be able to establish almost all of theglobal results in 8.1. However, there will still to be one final obstacle. It isthe case of a simple parameter P rsimpGq, which among other things, willbe essential for a resolution of our induction hypotheses. An examinationof this case leads us to the initial impression that it will be resistant to allof our earlier techniques. However, we will then see that there is a wayto treat it. We will introduce a second supplementary parameter, whichappears ungainly at first, but which, with the support of two rather intricatelemmas, takes us to a successful conclusion.

Section 8.2 is the climax of our long running induction argument, as wellas its most difficult point of application. Its final resolution is what bringsus to the end of the proof. We will then be free in 8.3 for some general

reflections that will give us some perspective on what has been established.In 8.4, we will sharpen our results for the groups SOp2nq, in which theouter automorphism creates some ambiguity. We will use the stabilizedtrace formula to construct the local and global L-packets predicted for thesegroups by the conjectural theory of endoscopy. In 8.5, we will describe anapproximation LF of the global Langlands group LF that is tailored to theclassical groups of this volume. It could potentially be used in place of thead hoc global parameters of 1.4 to streamline the statements of the globaltheorems.

We shall treat inner forms of orthogonal and symplectic groups in Chap-ter 9. The automorphic representation theory for inner twists is considerablyeasier for knowing what happens in the case of quasisplit groups. In par-ticular, the stable multiplicity formula is already in place, since it appliesonly to the quasisplit case. What work remains is primarily local. We shalldescribe how to modify our earlier arguments in order to classify local andglobal representations of arbitrary orthogonal and symplectic groups.

Having briefly summarized the various chapters, we had best add somecomment on our use of induction. As we have noted, induction is a central


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xvi PREFACE

part of the unified argument that will carry us from Chapter 2 to Section8.2. We will have two kinds of hypotheses, both based on the positive integer

N that indexes the underlying general linear group GLpNq. The first kindincludes various ad hoc assumptions, such as those implicit in some of ourdefinitions. For example, the global parameter sets rpGq defined in 1.4 arebased on the inductive application of two seed Theorems 1.4.1 and 1.4.2.The second will be the formal induction hypotheses introduced explicitly atthe beginning of 4.3, and in more refined form at the beginning of 5.1.They assert essentially that the stated theorems are all valid for parametersof rank less than N. In particular, they include the informal hypothesesimplicit in the definitions.

We do not actually have to regard the earlier, informal assumptionsas inductive. They really represent implicit appeals to stated theorems,in support of proofs of what amount to corollaries. In fact, from a logical

standpoint, it is simpler to treat them as inductive assumptions onlyafter weintroduce the formal induction hypotheses in 4.3 and 5.1. For a little morediscussion of this point, the reader can consult the two parallel Remarksfollowing Corollaries 4.1.3 and 4.2.4.

The induction hypotheses of 5.1 are formulated for an abstract familyrF of global parameters. They pertain to the parameters P rF of rankless than N, and are supplemented also by a hypothesis (Assumption 5.1.1)

for certain parameters in rF of rank equal to N. The results of Chapter5 will be applied three times, to three separate families rF. These are thefamily of generic parameters with local constraints used to establish the localclassification of Chapter 6, the family of nongeneric parameters with localconstraints used to deduce the local results for non-tempered representationsin Chapter 7, and the family of all global parameters used to establish theglobal theorems in the first two sections of Chapter 8. In each of these cases,the assumptions have to be resolved for the given family rF. In the case ofChapter 6, the induction hypotheses are actually imposed in two stages. Thelocal hypothesis at the beginning of 6.3 is needed to construct the familyrF, on which we then impose the global part of the general hypothesis ofChapter 5 at the beginning of 6.4. The earlier induction hypotheses of4.3 apply to general global parameters of rank less than N. They are usedin 4.34.6 to deduce the global theorems for parameters that are highlyreducible. Their general resolution comes only after the proof of the globaltheorems in 8.2.

Our induction assumptions have of course to be distinguished from thegeneral condition (Hypothesis 3.2.1) on which our results rely. This is thestabilization of the twisted trace formula for the two groups GLpNq andSOp2nq. As a part of the condition, we implicitly include twisted ana-logues of the two local results that have a role in the stabilization of theordinary trace formula. These are the orthogonality relations for elliptic


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PREFACE xvii

tempered characters of [A10, Theorem 6.1], and the weak spectral trans-fer of tempered p-adic characters given by [A11, Theorems 6.1 and 6.2].

The stabilization of orthogonality relations in 6.5, which requires twistedorthogonality relations for GLpNq, will be an essential part of the localclassification in Chapter 6. The two theorems from [A11] can be regardedas a partial generalization of the fundamental lemma for the full sphericalHecke algebra [Hal]. (Their global proof of course depends on the basicfundamental lemma for the unit, established by Ngo.) We will use them incombination with their twisted analogues in the proofs of Proposition 2.1.1and Corollary 6.7.4. The first of these gives the image of the twisted transferof functions from GLpNq, which is needed in the proof of Proposition 3.5.1.The second gives a relation among tempered characters, which completesthe local classification.

There is one other local theorem whose twisted analogues for GLpNq andSOp2nq we shall also have to take for granted. It is Shelstads strong spectraltransfer of tempered archimedean characters, which is to say, her endoscopicclassification of representations of real groups. This of course is major result.Together with its two twisted analogues, it gives the archimedean cases ofthe local classification in Theorems 2.2.1 and 2.2.4. We shall combine itwith a global argument in Chapter 6 to establish the p-adic form of thesetheorems. The general twisted form of Shelstads endoscopic classificationappears to be within reach. It is likely to be established soon by someextension of recent work by Mezo [Me] and Shelstad [S7].

Finally, let me include a word on the notation. Because our main theo-rems require interlocking proofs, which consume a good part of the volume,there is always the risk of losing ones way. Until the end of 8.2, assertions

as Theorems are generally stated with the understanding that their proofswill usually be taken up much later (unless of course they are simply quotedfrom some other source). On the other hand, assertions denoted Proposi-tions, Lemmasor Corollaries represent results along the route, for which thereader can expect a timely proof. The actual mathematical notation mightappear unconventional at times. I have tried to structure it so as to reflectimplicit symmetries in the various objects it represents. With luck, it mighthelp a reader navigate the arguments without necessarily being aware ofsuch symmetries.

The three main theorems of the volume were described in [A18, 30].I gave lecture courses on them in 19941995 at the Institute for AdvancedStudy and the University of Paris VII, and later in 2000, again at the Insti-

tute for Advanced Study. Parts of Chapter 4 were also treated heuristicallyin the earlier article [A9]. In writing this volume, I have added some top-ics to my original notes. These include the local Langlands classificationfor GLpNq, the treatment of inner twists in Chapter 9 and the remarks onWhittaker models in 8.3. I have also had to fill unforeseen gaps in thenotes. For example, I did not realize that twisted endoscopy for SOp2nqwas needed to formulate the local intertwining relation. In retrospect, it is


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probably for the best that this second case of twisted endoscopy does have arole here, since it forces us to confront a general phenomenon in a concrete

situation. I have tried to make this point explicit in 2.4 with the discussionsurrounding the short exact sequence (2.4.10). In any case, I hope that Ihave accounted for most of the recent work on the subject, in the referencesand the text. There will no doubt be omissions. I most regret not beingable to describe the results of Moeglin on the structure of p-adic packetsrv ([M1][M4]). It is clearly an important problem to establish analoguesof her results for archimedean packets.

Acknowledgment: I am indebted to the referee, for a close and carefulreading of an earlier draft of the manuscript (which went to the end of8.2), and for many very helpful comments.


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CHAPTER 1

Parameters

1.1. The automorphic Langlands group

We begin with some general motivation. We shall review some of thefundamental ideas that underlie the theoretical foundations laid by Lang-lands. This will help us put our theorems into perspective. It will also leadnaturally to a formulation of some of the essential objects with which we

need to work.We take F to be a local or global field of characteristic 0. In other

words, F is a finite extension of either the real field R or a p-adic field Qp,or it is a finite extension ofQ itself. Suppose that G is a connected reductivealgebraic group over F, which to be concrete we take to be a classical matrixgroup. For example, we could let G be the general linear group

GpNq GLpNqof invertible matrices of rank N over F.

In his original paper [L2], Langlands introduced what later becameknown as the L-group of G. This object is a semidirect product

LG pG Fof a complex dual group pG of G with the Galois group

F GalpF{Fqof an algebraic closure F ofF. The action of F on pG (called an L-action) isdetermined by its action on a based root datum for G and a corresponding

splitting for pG, according to the general theory of algebraic groups. (See[K1, 1.11.3].) It factors through the quotient E{F of F attached toany finite Galois extension E F over which G splits. We sometimesformulate the L-group by the simpler prescription

LG LGE{F pG E{F,since this suffices for many purposes. If G GpNq, for example, the actionof F on pG is trivial. Since pG is just the complex general linear groupGLpN, Cq in this case, one can often take

LG pG GLpN, Cq.1


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2 1. PARAMETERS

Langlands conjectures [L2] predicate a fundamental role for the L-groupin the representation theory of G. Among other things, Langlands conjec-

tured the existence of a natural correspondence

between two quite different kinds of objects. The domain consists of (con-tinuous) L-homomorphisms

: F LG,taken up to conjugation by pG. (An L-homomorphism between two groupsthat fibre over F is a homomorphism that commutes with the two projec-tions onto F.) The codomain consists of irreducible representations ofGpFq ifF is local, and automorphic representations ofGpAq ifF is global,taken in each case up to the usual relation of equivalence of irreducible rep-resentations.

Recall that if F is global, the adele ring is defined as a restricted tensorproduct

A AF v

Fv

of completions Fv of F. In this case, the Langlands correspondence shouldsatisfy the natural local-global compatibility condition. Namely, if v de-notes the restriction of to the subgroup Fv of F (which is defined up toconjugacy), and is a restricted tensor product

v v, v v,of representations that correspond to these localizations, then should cor-respond to . We refer the reader to the respective articles [F] and [L6] fora discussion of restricted direct products and automorphic representations.

The correspondence , which remains conjectural, is to be un-derstood in the literal sense of the word. For general G, it will not be amapping. However, in the case G GLpNq, the correspondence should infact reduce to a well defined, injective mapping. For local F, this is partof what has now been established. For global F, the injectivity would bea consequence of the required local-global compatibility condition and thetheorem of strong multiplicity one, or rather its generalization in [JS] thatwe will recall in 1.3. However, the correspondence will very definitely notbe surjective.

In our initial attempts at motivation, we should not lose sight of the factthat the conjectural Langlands correspondence is very deep. For example,even though the mapping is known to exist for G GLp1q, ittakes the form of the fundamental reciprocity laws of local and global classfield theory. Its generalization to GLpNq would amount to a formulation ofnonabelian class field theory.


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1.1. THE AUTOMORPHIC LANGLANDS GROUP 3

Langlands actually proposed the correspondence with the Weilgroup WF in place of the Galois group F. We recall that WF is a locally

compact group, which was defined separately for local and global F by Weil.It is equipped with a continuous homomorphism

WF F,with dense image. If F is global, there is a commutative diagram

WFv Fv

WF Ffor any completion Fv of F, with vertical embeddings defined up to conju-gation. (See [T2].) In the Weil form of the Langlands correspondence, represents an L-homomorphism from WF to

LG. The restriction mapping

of continuous functions on F to continuous functions on WF is injective.For this reason, the conjectural Langlands correspondence for Weil groupsis a generalization of its version for Galois groups.

For G GLpNq, the Weil form of the conjectural correspondence again reduces to an injective mapping. (In the global case, one has to take to be an isobaric automorphic representation, a natural restriction in-troduced in [L7] that includes all the representations in the automorphicspectral decomposition of GLpNq.) If G GLp1q, it also becomes surjec-tive. However, for nonabelian groups G, and in particular for GLpNq, thecorrespondence will again not be surjective. One of the purposes of Lang-lands article [L7] was to suggest the possibility of a larger group, whichwhen used in place of the Weil group, would give rise to a bijective mappingfor GLpNq. Langlands formulated the group as a complex, reductive, proal-gebraic group, in the spirit of the complex form of Grothendiecks motivicGalois group.

Kottwitz later pointed out that Langlands group ought to have an equiv-alent but simpler formulation as a locally compact group LF [K1]. It wouldcome with a surjective mapping

LF WFonto the Weil group, whose kernel KF should be compact and connected,and (in the optimistic view of some [A17]) even simply connected. If F islocal, LF would take the simple form

(1.1.1) LF #WF, if F is archimedean,

WF SUp2q, if F is nonarchimedean.In this case, LF is actually a split extension of WF by a compact, simplyconnected group (namely, the trivial group t1u if F is archimedean and thethree dimensional compact Lie group SUp2q SUp2,Rq ifF is p-adic.) IfFis global, LF remains hypothetical. Its existence is in fact one of the deepest


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4 1. PARAMETERS

problems in the subject. Whatever form it does ultimately take, it ought tofit into a larger commutative diagram

LFv WFv Fv

LF WF Ffor any completion Fv, the vertical embedding on the left again being definedup to conjugation.

The hypothetical formal structure of LF is thus compatible with anextension of the Langlands correspondence from WF to LF. This is whatLanglands proposed in [L7] (for the proalgebraic form ofLF). The extensionamounts to a hypothetical correspondence , in which now representsan L-homomorphism

: LF

LG,

taken again up to pG-conjugacy. Here it is convenient to use the Weil formof the L-group

LG pG WF,for the action ofWF on pG inherited from F. An L-homomorphism betweentwo groups over WF is again one that commutes with the two projections.There are some minor conditions on that are implicit here. For example,since WF and LF are no longer compact, one has to require that for any P LF, the image of pq in pG be semisimple. If G is not quasisplit, onegenerally also requires that be relevant to G, in the sense that if its imagelies in a parabolic subgroup LP of LG, there is a corresponding parabolicsubgroup P of G that is defined over F.

Suppose again that G GLpNq. Then the hypothetical extended cor-respondence again reduces to an injective mapping. However, thistime it should also be surjective (provided that for global F, we take theimage to be the set of isobaric automorphic representations). If F is localarchimedean, so that LF WF, the correspondence was established (forany G in fact) by Langlands [L11]. If F is a local p-adic field, so thatLF WF SUp2q, the correspondence was established for GLpNq by Har-ris and Taylor [HT] and Henniart [He1]. For global F, the correspondencefor GLpNq is much deeper, and remains highly conjectural. We have intro-duced it here as a model to motivate the form of the theorems we seek forclassical groups.

IfG is more general, the extended correspondence

will not reduceto a mapping. It was to account for this circumstance that Langlands intro-duced what are now called L-packets. We recall that L-packets are supposedto be the equivalence classes for a natural relation that is weaker than theusual notion of equivalence of irreducible representations. (The supplemen-tary relation is called L-equivalence, since it is arithmetic in nature, and isdesigned to preserve the L-functions and -factors of representations.) Theextended correspondence is supposed to project to a well defined


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1.1. THE AUTOMORPHIC LANGLANDS GROUP 5

mapping from the set of parameters to the set of L-packets. ForG GLpNq, L-equivalence reduces to ordinary equivalence. The L-packets then contain one element each, which is the reason that the correspon-dence reduces to a mapping in this case.

The general construction of L-packets is part of Langlands conjecturaltheory of endoscopy. It will be a central topic of investigation for this volume.We recall at this stage simply that the L-packet attached to a given willbe intimately related to the centralizer

(1.1.2) S Cent`

Impq, pGin pG of the image pLFq of , generally through its finite quotient(1.1.3) S S{S0Zp

pGq.

Following standard notation, we have written S0

for the connected compo-nent of 1 in the complex reductive group S, Zp pGq for the center of pG, andZp pGq for the subgroup of invariants in Zp pGq under the natural action ofthe Galois group F. For G GLpNq, the groups S have all to beconnected. Each quotient S is therefore trivial. The implication for othergroups G is that we will have to find a way to introduce the centralizers S,even though we have no hope of constructing the automorphic Langlandsgroup LF and the general parameters .

There is a further matter that must also be taken into consideration.Suppose for example that F is global and that G GLpNq. The problemin this case is that the conjectural parametrization of automorphic repre-sentations by N-dimensional representations

: LF pG GLpN, Cqis not compatible with the spectral decomposition of L2

`GpFqzGpAq. If

is irreducible, is supposed to be a cuspidal automorphic representation.Any such representation is part of the discrete spectrum (taken modulo thecenter). However, there are also noncuspidal automorphic representationsin the discrete spectrum. These come from residues of Eisenstein series,and include for example the trivial one-dimensional representation of GpAq.Such automorphic representations will correspond to certain reducible N-dimensional representations of LF. How is one to account for them?

The answer, it turns out, lies in the product of LF with the supplemen-

tary group SUp2q SUp2,Rq. The representations in the discrete auto-morphic spectrum of GLpNq should be attached to irreducible unitary N-dimensional representations of this product. The local constituents of theseautomorphic representations should again be determined by the restrictionof parameters, this time from the product LF SUp2q to its subgroupsLFv SUp2q. Notice that if v is a p-adic valuation, the localization

LFv SUp2q WFv SUp2q SUp2q


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6 1. PARAMETERS

contains two SUp2q-factors. Each will have its own distinct role. In 1.3, weshall recall the general construction, and why it is the product LF SUp2qthat governs the automorphic spectrum of GLpNq.Similar considerations should apply to a more general connected groupG over any F. One would consider L-homomorphisms

: LF SUp2q LG,with relatively compact image in pG (the analogue of the unitary conditionGLpNq). If F is global, the parameters should govern the automorphicspectrum of G. If F is local, they ought to determine corresponding lo-cal constituents. In either case, the relevant representations should occurin packets , which are larger and more complicated than L-packets, butwhich are better adapted to the spectral properties of automorphic repre-sentations. These packets should in turn be related to the centralizers

S Cent`Impq, pGand their quotients

S S{S0Zp pGq.For G GLpNq, the groups S remain connected. However, for otherclassical groups G we might wish to study, we must again be prepared tointroduce parameters and centralizers S without reference to the globalLanglands group LF.

The objects of study in this volume will be orthogonal and symplecticgroups G. Our general goal will be to classify the representations of suchgroups in terms of those of GLpNq. In the hypothetical setting of the dis-cussion above, the problem includes being able to relate the parameters for G with those for GLpNq. As further motivation for what is to come,we shall consider this question in the next section. We shall analyze theself-dual, finite dimensional representations of a general group F. Amongother things, this exercise will allow us to introduce endoscopic data, theinternal objects for G that drive the classification, in concrete terms.

1.2. Self-dual, finite dimensional representations

We continue to take F to be any local or global field of characteristic 0.For this section, we let F denote a general, unspecified topological group.The reader can take F to be one of the groups F, WF or LF discussedin 1.1, or perhaps the product of one of these groups with SU

p2q. We

assume only that F is equipped with a continuous mapping F F,with connected kernel and dense image.

We shall be looking at continuous, N-dimensional representations

r : F GLpN,Cq.Any such r factors through the preimage of a finite quotient of F. We cantherefore replace F by its preimage. In fact, one could simply take a large


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1.2. SELF-DUAL, FINITE DIMENSIONAL REPRESENTATIONS 7

finite quotient of F in place of F, which for the purposes of the presentexercise we could treat as an abstract finite group.

We say that r is self-dual if it is equivalent to its contragredient repre-sentationr_pq trpq1, P F,

where x tx is the usual transpose mapping. In other words, the equiva-lence class of r is invariant under the standard outer automorphism

pxq x_ tx1, x P GLpNq,of GLpNq. This condition depends only on the inner class of . It remainsthe same if is replaced by any conjugate

gpxq g1pxqg, g P GLpNq.We shall analyze the self-dual representations r in terms of orthogonal and

symplectic subgroups of GLpN, Cq.We decompose a given representation r into a direct sumr 1r1 rrr,

for inequivalent irreducible representations

rk : F GLpNk,Cq, 1 k r,and multiplicities k with

N 1N1 ` ` rNr.The representation is self-dual if and only if there is an involution k k_on the indices such that for any k, r_k is equivalent to rk_ and k k_. Weshall say that r is elliptic if it satisfies the further constraint that for each

k, k_ k and k 1. We shall concentrate on this case.Assume that r is elliptic. Then

r r1 rr,for distinct irreducible, self-dual representations ri of F of degree Ni. If iis any index, we can write

r_i pq AiripqA1i , P F,for a fixed element Ai P GLpNi,Cq. Applying the automorphism to eachside of this equation, we then see that

ripq A_i r_i pA_i q1 pA_i AiqripqpA_i Aiq1.

Since ri is irreducible, the product A_i Ai is a scalar matrix. We can thereforewrite

tAi ciAi, ci P C.If we take the transpose of each side of this equation, we see further thatc2i 1. Thus, ci equals `1 or 1, and the nonsingular matrix Ai is eithersymmetric or skew-symmetric. The mapping

x pA1i q tx Ai, x P GLpNq,


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8 1. PARAMETERS

of course represents the adjoint relative to the bilinear form defined by Ai.Therefore ripq belongs to the corresponding orthogonal group OpAi,Cq orsymplectic group SppAi,Cq, according to whether ci equals `1 or 1.Let us write IO and IS for the set of indices i such that ci equals `1 and1 respectively. We then write

rpq iPI

ripq, P F,

A iPI

Ai,

and

N iPI

Ni,

for equal to O or S. Thus AO is a symmetric matrix in GLpNO,Cq, AS isa skew-symmetric matrix in GLpNS,Cq, and rO and rS are representationsof F that take values in the respective groups OpAO,Cq and SppAS,Cq.We have established a canonical decomposition

r rO rSof the self-dual representation r into orthogonal and symplectic components.

It will be only the equivalence class of r that is relevant, so we are freeto replace rpq by its conjugate

B1rpqB

by a matrix B P GLpN, Cq. This has the effect of replacing the matrixA AO AS

by tBAB. In particular, we could take AO to be any symmetric matrix inGLpNO,Cq, and AS to be any skew-symmetric matrix in GLpNS,Cq. Wemay therefore put the orthogonal and symplectic groups that contain theimages of rO and rS into standard form.

It will be convenient to adopt a slightly different convention for thesegroups. As our standard orthogonal group in GLpNq, we take

OpNq OpN, Jq,where

J JpNq 0 1

. ..

1 0

is the second diagonal in GLpNq. This is a group of two connected com-ponents, whose identity component is the special orthogonal group

SOpNq x P OpNq : detpxq 1(.


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As the standard symplectic group in GLpNq, defined for N 2N1 even, wetake the connected group

SppNq SppN, J1qfor the skew-symmetric matrix

J1 J1pNq

0 JpN1qJpN1q 0

.

The advantage of this formalism is that the set of diagonal matrices ineither SOpNq or SppNq forms a maximal torus. Similarly, the set of uppertriangular matrices in either group forms a Borel subgroup. The point isthat if

tx Jtx J Jtx J1, x P GLpNq,denotes the transpose of x about the second diagonal, the automorphism

IntpJq : x J pxqJ1

tx1

of GLpNq stabilizes standard Borel subgroup of upper triangular matrices.Notice that there is a related automorphismrpNq Intp rJq : x rJ pxq rJ1defined by the matrix

(1.2.1) rJ rJpNq 0 11

. ..

p1qN 1 0

,which stabilizes the standard splitting in GL

pN

qas well. Both of these

automorphisms lie in the inner class of , and either one could have beenused originally in place of .

Returning to our discussion, we can arrange that A equalsJpNOq J1pNSq. It is best to work with the matrix

JO,S JpNO, NSq

0 JpN1SqJpNOq

JpN1Sq 0, NS 2N1S,

obtained from the obvious embedding of JpNOq J1pNSq into GLpN,Cq.The associated elliptic representation r from the given equivalence class thenmaps F to the corresponding subgroup of GLpN, Cq, namely the subgroup

O

pNO,C

q Sp

pNS,C

qdefined by the embedding

px, yq

y11 0 y120 x 0

y21 0 y22

,where yij are the four pN1S N1Sq-block components of the matrixy P SppNS,Cq.


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10 1. PARAMETERS

The symplectic part rS of r is the simpler of the two. Its image iscontained in the connected complex group

pGS SppNS,Cq.This in turn is the dual group of the split classical group

GS SOpNS ` 1q.The orthogonal part rO of r is complicated by fact that its image is

contained only in the disconnected group OpNO,Cq. Its composition withthe projection of OpNO,Cq onto the group

OpNO,Cq{SOpNO,Cq Z{2Zof components yields a character on F of order 1 or 2. Since we areassuming that the kernel of the mapping of F to F is connected, can

be identified with a character on the Galois group F of order 1 or 2. Thisin turn determines an extension E of F of degree 1 or 2.

Suppose first that NO is odd. In this case, the matrix pIq in OpNOqrepresents the nonidentity component, and the orthogonal group is a directproduct

OpNO,Cq SOpNO,Cq Z{2Z.We write

SOpNO,Cq pGO,where GO is the split group SppNO 1q over F. We then use to identifythe direct product

L

pG

qE{F

pGO E{Fwith a subgroup ofOpNO,Cq, namely SOpNO,Cq or OpNO,Cq, according towhether has order 1 or 2. We thus obtain an embedding of the (restricted)L-group of GO into GLpNO,Cq.

Assume next that N is even. In this case, the nonidentity component inOpNq acts by outer automorphism on SOpNOq. We write

SOpNO,Cq pGO,where GO is now the corresponding quasisplit orthogonal group SOpNO, qover F defined by . In other words, GO is the split group SOpNOq if istrivial, and the non-split group obtained by twisting SOpNOq over E by thegiven outer automorphism if is nontrivial. Let rwpNOq be the permutationmatrix in GLpNOq that interchanges the middle two coordinates, and leavesthe other coordinates invariant. We take this element as a representativeof the nonidentity component of OpNO,Cq. We then use to identify thesemidirect product

LpGOqE{F pGO E{Fwith a subgroup ofOpNO,Cq, namely SOpNO,Cq or OpNO,Cq as before. Weagain obtain an embedding of the (restricted) L-group ofGO into GLpNO,Cq.


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We have shown that the elliptic self-dual representation r factors throughthe embedded subgroup

LGE{F LpGOqE{F LpGSqE{Fof GLpN,Cq attached to a quasisplit group

G GO GSover F. The group G is called a -twisted endoscopic group for GLpNq. Itis determined by r, and in fact by the decomposition N NO ` NS and thecharacter G (of order 1 or 2) attached to r. The same is true of theL-embedding

O,S, : LG pG F LGpNq GLpN,Cq F,obtained by inflating the embedding above to the full L-groups.

It is convenient to form the semidirect productrG`pNq GLpNq xy GpNq xrpNqy,where xy and xrpNqy are the groups of order 2 generated by the automor-phisms and rpNq. We writerG0pNq GLpNq 1 GpNq 1for the identity component, which we can of course identify with the generallinear group GLpNq, and(1.2.2) rGpNq GLpNq GpNq rpNqfor the other connected component. Given r, and hence also the decompo-sition N

NO

`NS, we can form the semisimple element

s sO,S J 1O,S ,

in the dual setprGpNq of complex points GLpN,Cq. The complex grouppG pGO pGS, attached to r as above, is then the connected centralizer of

s in the group pGpNq prG0pNq GLpN,Cq.The triplet pG,s,q is called an endoscopic datum for rGpNq, since it becomesa special case of the terminology of [KS, p. 16] if we replace rGpNq with thepair

`r

G0pNq,

rpNq.

The endoscopic datum pG,s,q we have introduced has the property ofbeing elliptic. This is a consequence of our condition that the original self-dual representation r is elliptic. A general (nonelliptic) endoscopic datum

for rGpNq is again a triplet pG,s,q, where G is a quasisplit group over F, sis a semisimple element in

prGpNq of which pG is the connected centralizer inprG0pNq, and is an L-embedding of LG into the L-groups L rG0pNq LGpNqof GLpNq. (In the present setting, we are free to take either the Galois orWeil form of the L-groups.) We require that equal the identity on pG, and


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12 1. PARAMETERS

that the projection onto

prG0pNq of the image of lie in the full centralizer

of s. The endoscopic group G (or datum

pG,s,

q) for rGpNq then said to beelliptic ifZp pGq, the subgroup of elements in the center Zp pGq of pG invariant

under the action of the Galois group F, is finite.The notion of isomorphism between two general endoscopic data is de-

fined in [KS, p. 18]. In the case at hand, it is given by an element g in the

dual group pGpNq GLpN,Cq whose action by conjugation is compatible ina natural sense with the two endoscopic data. We writerAutNpGq Aut rGpNqpGqfor the group of isomorphisms of the endoscopic datum G to itself. Themain role for this group is in its image

rOutNpGq rAutNpGq{rIntNpGqin the group of outer automorphisms of the group G over F. (Followingstandard practice, we often let the endoscopic group G represent a full en-doscopic datum pG,s,q, or even an isomorphism class of such data.) If Grepresents one of the elliptic endoscopic data constructed above, rOutNpGq istrivial if the integer NO is odd or zero. In the remaining case that NO is evenand positive, rOutNpGq is a group of order 2, the nontrivial element beingthe outer automorphism induced by the nontrivial connected component ofOpNO,Cq.

We write

rEpNq E

rGpNq

for the set of isomorphism classes of endoscopic data for rGpNq, andrEellpNq Eell` rGpNqfor the subset of classes in rEpNq that are elliptic. The data pG,s,q, at-tached to equivalence classes of elliptic, self-dual representations r as above,form a complete set of representatives of rEellpNq. The set rEellpNq is thusparametrized by triplets pNO, NS, q, where NO ` NS N is a decomposi-tion ofN into nonnegative integers with NS even, and G is a characterof F of order 1 or 2 with the property that 1 if NO 0, and 1if NO 2. (The last constraint is required in order that the datum beelliptic.) The goal of this volume is to describe the representations of theclassical groups G in terms of those of GLpNq. The general arguments willbe inductive. For this reason, the case in which pG is either purely orthogonalor purely symplectic will have a special role. Accordingly, we writerEsimpNq Esim` rGpNqfor the set of elements in rEellpNq that are simple, in the sense that one ofthe integers NO or NS vanishes. We then have a chain of sets

(1.2.3) rEsimpNq rEellpNq rEpNq,


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which are all finite if F is local, and all infinite if F is global.The elements G P

rEpNq are usually called twisted endoscopic data, since

they are attached to the automorphism . We shall have to work also withordinary (untwisted) endoscopic data, at least for the quasisplit orthogonal

and symplectic groups G that represent elements in rEsimpNq. An endoscopicdatum G1 for G is similar to what we have described above for rGpNq. Itamounts to a triplet pG1, s1, 1q, where G1 is a (connected) quasisplit groupover F, s1 is a semisimple element in pG of which pG1 is the connected central-izer in pG, and 1 is an L-embedding of LG1 into LG. We again require that1 equal the identity on pG1, and that its image lie in the centralizer of s1 inLG. (See [LS1, (1.2)], a special case of the general definition in [KS], whichwe have specialized further to the case at hand.) There is again the notionof isomorphism of endoscopic data, which allows us to form the associatedfinite group

OutGpG1q AutGpG1q{IntGpG1qof outer automorphisms of any given G1. We write EpGq for the set ofisomorphism classes of endoscopic data G1 for G, and EellpGq for the subsetof data that are elliptic, in the sense that Zp pG1q is finite. We then have asecond chain of sets

(1.2.4) EsimpGq EellpGq EpGq,where EsimpGq tGu is the subset consisting of G alone. Similar definitionsapply to groups G that represent more general data in rEpNq.

It is easy to describe the set EellpGq, for any G P

rEsimpNq. It suffices to

consider diagonal matrices s1 P pG with eigenvalues 1. For example, in thefirst case that G SOpN ` 1q and pG SppN,Cq (with N NS even), itis enough to take diagonal matrices of the form

s1 I21 0

I120 I21

,where I21 is the identity matrix of rank N

21 N11{2, and I12 is the identity

matrix of rank N12. The set EellpGq is parametrized by pairs pN11, N12q ofeven integers with 0 N11 N12 and N N11 ` N12. The correspondingendoscopic groups are the split groups

G1 SOpN11 ` 1q SOpN12 ` 1q,

with dual groupspG1 SppN11,Cq SppN12,Cq SppN, Cq pG.The group OutGpG1q is trivial in this case unless N11 N12, in which case ithas order 2.

The other cases are similar. In the second case that G SppN 1qand pG SOpN, Cq (with N NO odd), EellpGq is parametrized by pairspN11, N12q of nonnegative even integers with N N11`pN12`1q, and characters


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14 1. PARAMETERS

1 on F with p1q2 1. The corresponding endoscopic groups are thequasisplit groups

G1

SOpN11,

1

q SppN12q,

with dual groupspG1 SOpN11,Cq SOpN12 ` 1,Cq SOpN, Cq pG.In the third case that G SOpN, q and pG SOpN,Cq (with N NOeven), EellpGq is parametrized by pairs of even integers pN11, N12q with0 N11 N12 and N N11 ` N12, and pairs p11, 12q of characters on Fwith p11q2 p12q2 1 and 1112. The corresponding endoscopic groupsare the quasisplit groups

G1 SOpN11, 11q SOpN12, 12q,with dual groupspG1 SOpN11,Cq SOpN12,Cq SOpN, Cq pG.In the second and third cases, each character 1 has to be nontrivial if thecorresponding integer N1 equals 2, while if N1 0, 1 must of course betrivial. In these cases, the group OutGpG1q has order 2 unless N is even andN11 N12 1, in which case it is a product of two groups of order 2, orN11 0, in which case the group is trivial.

Observe that in the even orthogonal case, where pG SOpN, Cq with Neven, the endoscopic data G1 P EellpGq will not be able to isolate constituentsri of r of odd dimension. The discrepancy is made up by a third kindof endoscopic datum. These are the twisted endoscopic data for the even

orthogonal groups G SOpN, q in rEsimpNq. For any such G, let(1.2.5) rG G r, r Int` rwpNq,be the nonidentity component in the semi-direct product of G with the

group of order two generated by the outer automorphism r of SOpNq. Inthis setting a (twisted) endoscopic datum is a triplet p rG1, rs1, r1q, where rG1is a quasisplit group over F, rs1 is a semisimple element in the dual setprG pG r of which prG1 is the connected centralizer in pG, and r1 is anL-embedding of L rG1 into LG, all being subject also to further conditionsand definitions as above. The subset Eellp

rGq Ep

rGq of isomorphism classes

of elliptic endoscopic data for

rG is parametrized by pairs of odd integers

p rN11, rN12q, with 0 rN11 rN12 and N rN11 ` rN12, and pairs of characterspr11, r12q on F, with pr11q2 pr12q2 1 and r11r12. The correspondingendoscopic groups are the quasisplit groupsrG1 Spp rN11q SppN12q,with dual groupsprG1 SOp rN11,Cq SOp rN12,Cq SOpN, Cq pG.


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1.3. REPRESENTATION OF GLpNq 15

The family Eellp

rGq will have to be part of our analysis. However, its role

will be subsidiary to that of the two primary families rEellpNq and EellpGq.We have completed our brief study of elliptic self-dual representations r.Remember that we are regarding these objects as parameters, in the spiritof1.1. We have seen that a parameter for GLpNq factors into a product oftwo parameters for quasisplit classical groups. The products are governed by

twisted endoscopic data G P rEellpNq. They can be refined further accordingto ordinary endoscopic data G1 P EellpGq. Thus, while the parameters are notavailable (for lack of a global Langlands group LF), the endoscopic data thatcontrol many of their properties are. Before we can study the ramificationsof this, we must first formulate a makeshift substitute for the parametersattached to our classical groups. We shall do so in 1.4, after a review in1.3 of the representations of GLpNq that will serve as parameters for thisgroup.

We have considered only the self dual representation r that are elliptic,since it is these objects that pertain directly to our theorems. Before goingon, we might ask what happens if r is not elliptic. A moments reflectionreveals that any such r factors through subgroups of GLpN,Cq attached toseveral data G P rEellpNq, in contrast to what we have seen in the ellipticcase. This is because r also factors through a subgroup attached to a datumM in the complement of rEellpNq in EpNq, and because any such M can beidentified with a proper Levi subgroup of several G. The analysis of generalself-dual representations r is therefore more complicated, though still notvery difficult. It is best formulated in terms of the centralizers

rSrpNq Sr` rGpNq Cent`Imprq, prGpNqandSr SrpGq Cent

`Imprq, pGq

of the images of r. We shall return to this matter briefly in 1.4, and thenmore systematically as part of the general theory of Chapter 4.

1.3. Representations of GLpNqA general goal, for the present volume and beyond, is to classify rep-

resentations of a broad class of groups in terms of those of general lineargroups. What makes this useful is the fact that much of the representationtheory of GLpNq is both well understood and relatively simple. We shallreview what we need of the theory.Suppose first that F is local. In this case, we can replace the abstractgroup F of the last section by the local Langlands group LF defined by(1.1.1). The local Langlands classification parametrizes irreducible repre-sentations of GLpN, Fq in terms of N-dimensional representations

: LF GLpN, Cq.Before we state it formally, we should recall a few basic notions.


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16 1. PARAMETERS

Given a finite dimensional (semisimple, continuous) representation ofLF, we can form the local L-function Lps, q, a meromorphic function ofs P C. We can also form the local -factor ps,,Fq, a monomial of theform abs which also depends on a nontrivial additive character F of F.If F is archimedean, we refer the reader to the definition in [T2, 3]. If Fis p-adic, we extend analytically to a representation of the product of WFwith the complexification SLp2,Cq of the subgroup SUp2q of LF. We canthen form the representation

pwq

w,

|w| 12 0

0 |w| 12

, w P WF,

of WF, where |w| is the absolute value on WF, and the nilpotent matrix

N

log 1,

1 1

0 1 .The pair V p, Nq gives a representation of the Weil-Deligne group[T2, (4.1.3)], for which we define an L-function

Lps, q ZpV, qsF qand -factor

ps,,Fq pV, qsF q,following notation in [T2, 4]. (We have written qF here for the order of theresidue field of F.) Of particular interest are the tensor product L-function

Lps, 1 2q Lps, 1 b2qand -factor

ps, 1 2, Fq ps, 1 b2, Fq,attached to any pair of representations 1 and 2 of LF.

We expect also to be able to attach local L-functions Lps , , rq and -factors ps,,r,Fq to any connected reductive group G over F, where ranges over irreducible representations ofGpFq, and r is a finite dimensionalrepresentation of LG. For general G, this has been done in only the sim-plest of cases. However, if G is a product GpN1q GpN2q of general lineargroups, there is a broader theory [JPS]. (See also [MW2, Appendice].) Itapplies to any representation 1 2, in the case that r is the standardrepresentation

(1.3.1) rpg1, g2q : X g1 X tg2, gi P GpNiq,of pG GLpN1,Cq GLpN2,Cqon the space of complex pN1 N2q-matrices X. The theory yields functions

Lps, 1 2q Lps , , rqand

ps, 1 2, Fq ps,,r,Fq,


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known as local Rankin-Selberg convolutions.The local classification for GLpNq is essentially characterized by being

compatible with local Rankin-Selberg convolutions. It has other importantproperties as well. Some of these relate to supplementary conditions we canimpose on the parameters as follows.

Suppose for a moment that G is any connected group over F. Wewrite pGq for the set of pG-orbits of (semisimple, continuous, G-relevant)L-homomorphisms

: LF LG,and pGq for the set of equivalence classes of irreducible (admissible) rep-resentations of GpFq. (See [Bo].) These sets come with parallel chains ofsubsets

2,bdd

pG

q bdd

pG

q

pG

qand2,temppGq temppGq pGq.

In the second chain, temppGq denotes the set of tempered representationsin pGq, and

2,temppGq 2pGq X temppGq,where 2pGq is the set of representations in pGq that are essentially squareintegrable, in the sense that after tensoring with the appropriate positivecharacter on GpFq, they are square integrable modulo the centre of GpFq.Recall that temppGq can be described informally as the set of represen-tations

P

pG

qthat occur in the spectral decomposition of L2`GpFq.Similarly, 2,temppGq is the set of that occur in the discrete spectrum

(taken modulo the center). In the first chain, bddpGq denotes the set of P pGq whose image in LG projects onto a relatively compact subset ofpG, and

2,bddpGq 2pGq X bddpGq,where 2pGq is the set of parameters in pGq whose image does not lie inany proper parabolic subgroup LP of LG.

In the case G GpNq GLpNq of present concern, we writepNq `GLpNq and pNq `GLpNq, and follow similar notationfor the corresponding subsets above. Then pNq can be identified with theset of equivalence classes of (semisimple, continuous) N-dimensional repre-

sentations of LF. The subset

simpNq 2pNqconsists of those representations that are irreducible, while bddpNq cor-responds to representations that are unitary. On the other hand, the setunitpNq of unitary representations in pNq properly contains temppNq, ifN 2. It has an elegant classification [V2], [Tad1], but our point here is


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18 1. PARAMETERS

that it is not parallel to the set ofN-dimensional representations that areunitary. We do observe that

2,temppNq 2pNq X unitpNq 2,unitpNq,so the notions of tempered and unitary are the same for square integrablerepresentations.

If F is p-adic, we can write scusp,temppNq (resp. scusppNq) for theset of supercuspidal representations in 2,temppNq (resp. 2pNq). We canalso write scusp,bddpNq (resp. scusppNq) for the set of in sim,bddpNq(resp. simpNq) that are trivial on the second factor SUp2q of LF. If Fis archimedean, it is natural to take scusp,temppNq and scusppNq to beempty unless GLpN, Fq is compact modulo the center (which is a silly wayof saying that N 1), in which case we can take them to be the correspond-ing sets 2,temp`GLp1q temp`GLp1q and 2`GLp1q `GLp1q. Wethus have two parallel chains(1.3.2) scusp,bddpNq sim,bddpNq bddpNq pNq,and

(1.3.3) scusp,temppNq 2,temppNq temppNq pNq,for our given local field F.

The local classification for G GLpNq can now be formulated as follows.Theorem 1.3.1 (Langlands [L11], Harris-Taylor [HT], Henniart [He1]).

There is a unique bijective correspondence from pNq onto pNqsuch that

piq b b p detq,for any character in the group p1q p1q,piiq det ,for the central character of , and

piiiq _ _,for the duality involutions _ on pNq and pNq, and such that if

i i, i P pNiq, i 1, 2,then

pivq Lps, 1 2q Lps, 1 2qand

pvq ps, 1 2, Fq ps, 1 2, Fq.Furthermore, the bijection is compatible with the two chains (1.3.2) and(1.3.3), in the sense that it maps each subset in (1.3.2) onto its counterpartin (1.3.3).


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We now take F to be global. This brings us to the representations thatwill be the foundation of all that follows. They are the objects in the set

AcusppNq Acusp`GLpNqof (equivalence classes of) unitary, cuspidal automorphic representations ofGLpNq.

Assume first that G is an arbitrary connected group over the global fieldF. To suppress the noncompact part of the center, one often works with theclosed subgroup

GpAq1 x P GpAq : |pxq| 1, P XpGqF(of GpAq, where XpGqF is the additive group of characters of G definedover F. We recall that GpFqzGpAq1 has finite volume, and that there is asequence

L2

cusp`GpFqzGpAq1 L2disc`GpFqzGpAq1 L2`GpFqzGpAq1,of embedded, right GpAq1-invariant Hilbert spaces. In particular, the spaceL2cusp

`GpFqzGpAq1 of cuspidal functions in L2`GpFqzGpAq1 is contained in

the subspace L2disc`

GpFqzGpAq1 that decomposes under the action ofGpAq1into a direct sum of irreducible representations. We can then introduce acorresponding chain of subsets of irreducible automorphic representations

AcusppGq A2pGq ApGq.By definition, AcusppGq, A2pGq and ApGq denote the subsets of irreducibleunitaryrepresentations ofGpAq whose restrictions to GpAq1 are irreducibleconstituents of the respective spaces L2cusp, L

2disc and L

2. We shall also

writeA`

cusppGq andA`

2 pGq for the analogues ofA

cusppGq andA

2pGq definedwithout the condition that be unitary. (The definition ofApGq here issomewhat informal, and will be used only for guidance. It can in fact bemade precise at the singular points in the continuous spectrum where theremight be some ambiguity.)

We specialize again to the case G GLpNq, taken now over the globalfield F. We write ApNq A`GLpNq, with similar notation for the cor-responding subsets above. Observe that GLpN, Aq1 is the group of adelicmatrices x P GLpN,Aq whose determinant has absolute value 1. In thiscase, the set ApNq is easy to characterize. It is composed of the inducedrepresentations

IP

p1

b b r

q, i

PA2

pNi

q,

where P is the standard parabolic subgroup of block upper triangular ma-trices in G GLpNq corresponding to a partition pN1, . . . , N rq of N, withthe standard Levi subgroup of block diagonal matrices

MP GLpN1q GLpNrq.This follows from the theory of Eisenstein series [L1], [L5], [A1] (validfor any G), and the fact [Be] (special to G GLpNq) that an induced


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20 1. PARAMETERS

representation IGPpq is irreducible for any representation of MPpAq thatis irreducible and unitary. Let us also write A`pNq for the set of inducedrepresentations as above, but with the components i now taken fromthe larger sets A`2 pNiq. These representations can be reducible at certainpoints, although they typically remain irreducible. We thus have a chain ofsets

(1.3.4) AcusppNq A2pNq ApNq A`pNqof representations, for our given global field F. This is a rough globalanalogue of the local sequence (1.3.3). We have dropped the subscripttemp in the global notation, since the local components of a constituent ofL2

`GLpN, FqzGLpN,Aq1 need not be tempered, and added the superscript

` to remind ourselves that A`pNq contains full induced representations,rather than irreducible quotients.

There are two fundamental theorems on the automorphic representationsof GLpNq that will be essential to us. The first is the classification ofautomorphic representations by Jacquet and Shalika, while the second isthe characterization by Moeglin and Waldspurger of the discrete spectrumin terms of cuspidal spectra. We shall review each in turn.

Suppose again that G is an arbitrary connected group over the globalfield F. An automorphic representation of G is among other things, aweakly continuous, irreducible representation of GpAq. As such, it can bewritten as a restricted tensor product

v

v

of irreducible representations of the local groups Gp

Fvq

, almost all whichare unramified [F]. Recall that if v is unramified, the group Gv G{Fv isunramified. This means that Fv is a p-adic field, that Gv is quasisplit, andthat the action of WFv on pG factors through the infinite cyclic quotient

WFv{IFv xFrobvyof WFv by the inertia subgroup IFv with canonical generator Frobv. Recallalso that the local Langlands correspondence has long existed in this veryparticular context. The pG-orbit of homomorphisms

v : LFv LGv pG xFrobvyin pGvq to which v corresponds factors through the quotient

LFv{`IFv SUp2q WFv{IFvof LFv . The resulting mapping

v cpvq vpFrobvqis a bijection from the set of unramified representations of GpFvq (relativeto any given hyperspecial maximal compact subgroup Kv GpFvq) and theset of semisimple pG-orbits in LGv that project to the Frobenius generator


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of WFv{IFv . Let cvpq be the image of cpvq in LG under the embeddingof LGv into

LG that is defined canonically up to conjugation. In this way,

the automorphic representation of G gives rise to a family of semisimpleconjugacy classes

cSpq cvpq : v R S(in LG, where S is some finite set of valuations of F outside of which G isunramified.

Let CSautpGq be the set of familiescS tcv : v R Su

of semisimple conjugacy classes in LG obtained in this way. That is,cS cSpq, for some automorphic representation of G. We define CautpGqto be the set of equivalence classes of such families, cS and pc1qS1 beingequivalent if cv equals c

1

v

for almost all v. We then have a mapping

cpqfrom the set of automorphic representations of G onto CautpGq.

The families cSpq arise most often in the guise of (partial) global L-functions. Suppose that r is a finite dimensional representation of LG thatis unramified outside ofS. The corresponding incomplete L-function is givenby an infinite product

LSps , , rq vRS

Lps, v, rvq

of unramified local L-functions

Lps, v, rvq det `1 r`cpvqqsv 1, v R S,which converges for the real part of s large. It is at the ramified places v P Sthat the construction of local L-functions and -factors poses a challenge.As the reader is no doubt well aware, the completed L-function

Lps , , rq v

Lps, v, rvq

is expected to have analytic continuation as a meromorphic function of s P Cthat satisfies the functional equation

(1.3.5) Lps,,rq ps , , rqLp1 s , , r_q,where r_ is the contragredient of r, and

ps , , rq vRS

ps, v, rv, Fvq.

Here, Fv stands for the localization of a nontrivial additive character Fon A{F that is unramified outside of S.

We return again to the case G GLpNq. The elements in the setCautpNq Caut

`GLpNq


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22 1. PARAMETERS

can be identified with families of semisimple conjugacy classes in GLpN,Cq,defined up to the equivalence relation above. It is convenient to restrict the

domain of the corresponding mapping cpq.Recall that general automorphic representations for GLpNq can be char-acterized as the irreducible constituents of induced representations

IPp1 b b rq, i A`cusppNiq.(See Proposition 1.2 of [L7], which applies to any G.) With this condi-tion, the nonunitary induced representation can have many irreducibleconstituents. However, it does have a canonical constituent. This is theirreducible representation

v

v,

where v P `

GLpNqv

is obtained from the local Langlands parameter

v 1,v r,v, i,v i,v,in

`GLpNqv

by Theorem 1.3.1 (applied to both GLpNqv and its subgroups

GLpNiqv). Isobaric representations are the automorphic representations of GLpNq obtained in this way. The equivalence class of does not changeif we reorder the cuspidal representations tiu. We formalize this propertyby writing

(1.3.6) 1 r, i P A`cusppNiq,in the notation [L7, 2] of Langlands, where the right hand side is regardedas a formal, unordered direct sum.

Theorem 1.3.2 (Jacquet-Shalika [JS]). The mapping

1 r cpq,from the set of equivalence classes of isobaric automorphic representations of GLpNq to the set of elements c cpq in CautpNq, is a bijection.

Global Rankin-Selberg L-functions Lps, 1 2q and -factorsps, 1 2q are defined as in the local case. They correspond to auto-morphic representations 1 2 of a group G GLpN1q GLpN2q,and the standard representation (1.3.1) of pG. The analytic behaviour ofthese functions is now quite well understood [JPS], [MW2, Appendice].In particular, Lps, 1 2q has analytic continuation with functional equa-tion (1.3.5). Moreover, if 1 and 2 are cuspidal, Lps, 1 2q is an entirefunction ofs unless N1 equals N2, and

_

2 is equivalent to the representation1pg1q | det g1|s11, g1 P GLpN1,Aq,

for some s1 P C, in which case Lps, 1 2q has only a simple pole at s s1.It is this property that is used to prove Theorem 1.3.2.

Theorem 1.3.2 can be regarded as a characterization of (isobaric) au-tomorphic representations of GLpNq in terms of simpler objects, familiesof semisimple conjugacy classes. However, it does not in itself characterize


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the spectral properties of these representations. For example, it is not apriori clear that the representations in ApNq, or even its subset A2pNq, areisobaric. The following theorem provides the necessary corroboration.

Theorem 1.3.3 (Moeglin-Waldspurger [MW2]). The representations P A2pNq are parametrized by the set of pairs pm, q, where N mn isdivisible by m, and is a representation inAcusppmq. If P is the standardparabolic subgroup corresponding to the partition pm , . . . , mq of N, and is the representation

x px1q| det x1|n12 bpx2q| det x2|

n32 b b pxnq| det xn|

pn1q2

of the Levi subgroup

MPpAq

x px1, . . . , xnq : xi P GLpm,Aq(

,

then is the unique irreducible quotient of the induced representation

IPpq. Moreover, the restriction of to GLpN,Aq1 occurs in the discretespectrum with multiplicity one.

Consider the representation P A2pNq described in the theorem. Forany valuation v, its local component v is the Langlands quotient of thelocal component IPp,vq of IPpq. It then follows from the definitionsthat v is the irreducible representation of GLpN, Fvq that corresponds tothe Langlands parameter of the induced representation IPp,vq. In otherwords, the automorphic representation is isobaric. We can therefore write

(1.3.7)

n12

n3

2

pn1q

2

,

in the notation (1.3.6), for cuspidal automorphic representations

piq : xi pxq|x|i, x P GLpm,Aq.Theorem 1.3.3 also provides a description of the automorphic represen-

tations in the larger set ApNq. For as we noted above, ApNq consists of theset of irreducible induced representations

IPp1 b b rq, i A2pNiq,where N N1 ` ` Nr is again a partition of N, and P is the corre-sponding standard parabolic subgroup of GLpNq. It is easy to see from itsirreducibility that is also isobaric. We can therefore write

(1.3.8) 1 r, i A2pNiq.Observe that despite the notation, (1.3.8) differs slightly from the general

isobaric representation (1.3.6). Its constituents i are more complex, sincethey are not cuspidal, but the associated induced representation is simpler,since it is irreducible.

There is another way to view Theorem 1.3.3. Since F is global, theLanglands group LF is not available to us. If it were, we would expect itsset pNq of (equivalence classes of) N-dimensional representations

: LF GLpN, Cq


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24 1. PARAMETERS

to parametrize the isobaric automorphic representations of GLpNq. A verysimple case, for example, is the pullback || to LF of the absolute value onthe quotient

WabF GLp1, FqzGLp1,Aqof LF. This of course corresponds to the automorphic character on GLp1qgiven by the original absolute value. We would expect the subset

sim,bddpNq 2,bddpNqof irreducible unitary representations in pNq to parametrize the unitarycuspidal automorphic representations of GLpNq. How then are we to ac-count for the full discrete spectrum A2pNq? A convenient way to do so isto take the product of LF with the group SUp2q SUp2,Rq.

Let us write pNq temporarily for the set of (equivalence classes of)unitary N-dimension representations

: LF SUp2q GLpN, Cq.According to Theorem 1.3.3, it would then be the subset

simpNq 2pNqof irreducible representations in pNq that parametrizes A2pNq. Indeed,any P simpNq decomposes uniquely as a tensor product b of irre-ducible representations of LF and SUp2q. It therefore gives rise to a pairpm, q, in which P Acusppmq represents a unitary cuspidal automorphicrepresentation of GLpNq as well as the corresponding irreducible, unitary,m-dimensional representation of LF. Conversely, given any such pair, we

again identify PA

cusppmq with the corresponding representation of LF,and we take to be the unique irreducible representation ofSUp2q of degreen N m1.

We are identifying any finite dimensional representation of SUp2q withits analytic extension to SLp2,Cq. With this convention, we attach an N-dimensional representation

: u

u,

|u| 12 0

0 |u|12

, u P LF,

of LF to any P pNq. If b belongs to the subset simpNq, decomposes as a direct sum

u puq|u|n1

2 puq|u|pn1

2 q,

to which we associate the induced representation IPpq of Theoreom 1.3.3.According to the rules of the hypothetical global correspondence from pNqto isobaric automorphic representations of GLpNq, it is the unique irre-ducible quotient of IPpq that is supposed to correspond to theparameter . This representation is unitary, as of course is implicit inTheorem 1.3.3. The mapping is thus an explicit realization of the


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bijective correspondence from simpNq to A2pNq implied by the theorem ofMoeglin and Waldspurger (and the existence of the group LF).

More generally, suppose that belongs to the larger set pNq. Thenthe isobaric representation attached to the parameter P pNq belongsto ApNq. It can be described in the familiar way as the irreducible unitaryrepresentation induced from a unitary representation of a Levi subgroup.The mapping becomes a bijection from pNq to ApNq. In general,the restriction of parameters is an injection from pNq into pNq.The role of the set pNq we have just defined in terms of the supplementarygroup SUp2q is thus to single out the subset of pNq that corresponds tothe subset ApNq of globally tempered automorphic representations.

We are also free to form larger sets

`pNq pNq

and`simpNq simpNq

of representations of LF SUp2q by removing the condition that they beunitary. Any element P `pNq is then a direct sum of irreducible rep-resentations i P `simpNiq. The components i should correspond to au-tomorphic representations i i in A`2 pNiq. The corresponding inducedrepresentation

IPp1 b b rqthen belongs to the set ofA`pNq of (possibly reducible) representations ofGLpN, Aq introduced above. Notice that the extended mapping from `pNq to pNq is no longer injective. In particular, will not ingeneral be equal to the automorphic representation corresponding to .However, the mapping will be a bijection from `pNq to A`pNq.In the interests of symmetry, we can also write

cusppNq sim,bddpNqfor the set of representations that are trivial o

arthur book endo

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