Arthur's Invariant Trace Formula and Comparison of Inner Forms

573 Pages·2016·7.01 MB·English

by  Yuval Z. Flicker (auth.)

Yuval Z. Flicker
Arthur’s Invariant
Trace Formula
and Comparison
of Inner Forms

Yuval Z. Flicker
Arthur’s Invariant Trace
Formula and Comparison
of Inner Forms
YuvalZ.Flicker
ArielUniversity
Ariel,Israel
TheOhioStateUniversity
Columbus,Ohio,USA
ISBN978-3-319-31591-1 ISBN978-3-319-31593-5 (eBook)
DOI10.1007/978-3-319-31593-5
LibraryofCongressControlNumber:2016940094
MathematicsSubjectClassification(2010):11F70,11F72,11F41,11M36,22E55,22E57
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Preface
The theory of automorphic representations of the group G.A/ of the adèle points
of a reductive connected group G over a global field F, and that of admissible
representations of the group G.Fv/ of points of a reductive connected group G
over a local field Fv, are governed by a hypothetical reciprocity law, introduced
byLanglands,thatrelatesthemto representationsof avariantoftheGaloisgroup
of the base field, namedWeil or Weil-Deligne group,into the complexLanglands
dualgroupLGofG.
This “principle of functoriality”—not touched upon in the present tome—
suggests relations between such automorphic and admissible representations of
different groups G. These relations have been termed liftings, correspondences,
transfers,andaresuggestedbyrelationsamongsttheunderlyingdualgroups.
For example, establishing lifting from GL.2/ to GL.n C 1/ corresponding to
the irreducible n-dimensional representation Symn from the dual group GL.2;C/
toGL.nC1;C/wouldimplytheRamanujanconjectureforGL.2/.Someofthese
liftings,which are analyticimplicationsof the principle,have beenestablished by
various techniques, using various invariants of the representations. The work of
Jacquet-Langlands[JL70]showedthattheSelbergtraceformula[Se62]couldgive
very complete results on the correspondence of representations of GL.2/ and its
innerforms,themultiplicativegroupsofquaternionalgebras.
Deligne and Kazhdan [DK] then introduced a simple form of the trace for-
mula, which applies to test functions—and representations—with two cuspidal
components (“two” was reduced later to “one” in a form of the simple trace
formuladevelopedby Flicker and Kazhdan [FK88]), and established in [DKV84]
the correspondence of representations between GL.n/ and its inner forms, the
multiplicativegroupsofsimplealgebras.
This work played the global trace formula against local analysis and used
multipleinductionargumentstoprovenotonlytheliftingofrepresentationsbutalso
theexistenceofmatchingorbitalintegrals.Thelatterwaspreviouslyconsideredto
beaprerequisiteforderivingliftingapplicationsfromthetraceformula.
To remove this last constraint, and of course for other applications, Arthur
developed the trace formula for a general test function, in a series of papers,
v
vi Preface
over many years. In particular, Arthur put the trace formula in invariant form,
namely, expressed all terms that appear in the formula as invariant distributions.
Thisis necessaryforcomparisonapplications,aswhencomparingrepresentations
of different groups, only characters or orbital integrals can be related, and this is
through a norm map relating conjugacyclasses of elements in the two groups. In
crudeterms, onlyeigenvaluesofelementscan be relatedbetweentwo groups,not
individualelements.
The presentvolumegrew outof an attemptto study Arthur’swork. We started
inacoursefollowingArthur’sexpositorynotes[Ar05],butquicklyrealizedthatto
attemptto understandthe theorywe had to study the source articles. To make the
subject more accessible, we decided to unite the main articles in one volume and
rewritethemasoneunitina conventionalway.Thuswe cutmanyargumentsinto
lemmas, propositions, and theorems, stated the claims before giving their proofs,
uniformizedthenotationtomakeiteasiertoread(e.g.,(cid:2) onGbut(cid:3) onM),andthe
like.
Thus, in Chapter 3, we explain Arthur’s proof of the basic, noninvarianttrace
formula,followinghisearlyDukeandCompositiopapers.
InChapter4,weexplainArthur’sAnnalsandJ.Funct.Anal.papers[Ar81]and
[Ar89], whichstudy the noninvarianceof the termsin the basic trace formulaand
preparetheingredientsforsettinguptheinvariantformula.
In Chapter 5, we explain Arthur’s JAMS papers [Ar88.2, Ar88.3], where the
invariant formula is finally developed. We quote some of Arthur’s computations
ofcontributionstothecontinuousspectrumfromAmer.J.Math.,[Ar82.I,Ar82.II],
andofweightedorbitalintegralsfromDuke[Ar88.1]and[Ar85,Ar86],wherewe
feltwecouldnotimprovetheexpositionsufficientlytojustifytheincreasedvolume.
Thus Chapters 3–5 here give an almost complete attempt to develop the invariant
traceformulainaformfitforapplications.
Toillustratetheuseofthistraceformula,inChapter6,wecomparetheinvariant
traceformulaeforG0 D GL.n/anditsinnerformG,formatchingfunctionsf0 on
G0.A/andf onG.A/,thusfunctionswithmatchingorbitalintegrals.Thisisalready
containedin[AC89,Chapter2],asthesecondarycase,accompanyingthemaincase
ofinterestthere:basechangeforGL.n/.Thismarriagemakesithardtofollowthe
innerformscase,sowedecidedtowriteitseparatelyinourChapter6.Thisisafter
all the initial case of comparison; in principle it should be the simplest, and we
thoughtconsequentlythatitdeserveditsownfulltreatment.
Thecomparisonofthetwotraceformulaeisfarfrombeingsimple.Akeyargu-
ment is a multiple induction process, reminiscent of Kazhdan’s double induction
argumentusingthesimpletraceformula.
Equipped with the comparison from Chapter 6, we set in Chapter 2 to prove
the correspondencebetween GL.n/ andits innerformsin general.This chapteris
basedonourcourse[F87.1]atHarvard1986,whereadifferentproofoftheresults
of [DKV84] was given, based on using the then recent works [BDK86] on the
trace Paley-Wiener theorem of Bernstein, Deligne, and Kazhdan, and on cuspidal
geometry[Ka86.1]byKazhdan.
Preface vii
The first section of Chapter 2 deals with the comparison of GL.n/ and the
anisotropic inner forms, multiplicative groups of a division algebra, following
[F90.2], using the simple trace formula.This is the only“easy” case, so we bring
it first. It was once considered difficult too, but [F90.2] observed that a finiteness
resultknownat the time permitteda relativelyeasy proof.Thisfinitenessresultis
nowknownforallinnerformsofGL.n/,butisnotneededintheproofofthegeneral
case.
The second section of Chapter 2 discusses the asymptotic behavior of orbital
integrals, following Shalika [Shal72], who used ideas of Harish-Chandra[HC70].
Aningredienthereistheconvergenceoftheorbitalintegralsontheunipotentorbits
in G. There is a publication of Rao [Ra72] on this, and we sketch, toward the
end of this Section 2, a proof of Deligne of this fact. This section ends with an
elegantcomputationofKottwitz[Ko88]oftheorbitalintegraloftheEuler-Poincaré
functionheintroduced,whichgivesexplicitlyapseudo-coefficientoftheSteinberg
representation.
In Section 24, we extend the correspondence from the non-degenerate case
of square-integrable, tempered, standard, or relevant local representations—what
followsonusingpurelythesimpletraceformulainSection13,toacorrespondence
of local unitarizable representations from GL.n/ to its inner forms, by purely
local arguments of the type of [BZ76, BZ77, Ze80, Tc90], and the Langlands
classification[BW80],followingBadulescu[Ba08].
Section 25 uses the comparison of the invariant trace formulae of Chapter 6
to establish the full global correspondence, for cuspidal representations without
any local constraint and also for residual representations, permitting to transfer
multiplicityoneandrigiditytheoremsfromtheknowncaseofthesplitgroupGL.n/
tothatoftheinnerforms,wherenoglobalrepresentationisgeneric;thusthetheory
of Whittaker modelsis notavailable, as well as establishing for the inner formsa
descriptionoftheresidualspectrum,analogoustothatestablishedbyMoeglinand
Waldspurger[MW89]forGL.n/.
WepostponetothefinalSection26ofChapter2anaccountofthesimpletrace
formula of Flicker and Kazhdan [FK88] that uses one cuspidal component and a
secondcomponentregular—thatleadstonoconstraintsonliftingapplications.The
ideaofregularfunctionswasinspiredbyDeligne’sconjectureonthevalidityofthe
Lefschetzfixedpointformulafora correspondenceonavarietyoverafinitefield,
provideditissufficientlytwistedbytheFrobenius;see[F13,Fu97,Va07].
Chapter 2 was used for a course at Ohio State in 2014. Chapter 1 contains a
statement of the results of Chapter 2 on the correspondence between GL.n/ and
its inner forms, as well as a summary of the statement of Arthur’s invarianttrace
formula. Since the statement is so involved, it is not surprising that the proof
and the development of the trace formula are so long. We hope this work would
make Arthur’s work more accessible. We note, however, that it, and in fact the
entire theory of automorphicrepresentations,is based on the theory of Eisenstein
series,ofLanglands[La66,La76];seealso[Ar79],bestexplainedbyMoeglinand
Waldspurger[MW95],thatremainsafundamentalchallenge.
viii Preface
This work was partially supported by grants from the Simons Foundation
(#267097 and #317731 to Yuval Z. Flicker), as well as the Humboldt-Stiftung,
TÜBITAK, and ERC AdG Grant 247049, while the author was a Schonbrunn
professor at the Hebrew University. The author enjoyed discussions with and
hospitalityofElmarGroße-KlönneatHUBerlin,DavidKazhdanatHUJerusalem,
Takayuki Oda at Tokyo U., Eric Opdam at Amsterdam, and Engin Özkan at
Erzincan,duringthe preparationof thisvolume.Thisworkgreatlybenefitedfrom
thecollaborationofMichaelBelfantiatOhioStatewhoattendedthecourses,took
notes,improvedtheexposition,andwasinvitedtocoauthorthisbook,butdeclined.
The envisaged readership of this book consists of graduate students and
researchersinterestedinthetraceformulaanditsapplications,especiallytolifting
problems.We hopeit wouldsimplify—infactmakeit possiblefor suchpeopleto
enterthissubject.
Ariel,Israel YuvalZ.Flicker
Contents
Preface............................................................................. v
1 Introduction .................................................................. 1
1 Motivation................................................................ 1
2 CorrespondenceBetweenGL.n/andItsInnerForms................. 4
3 Arthur’sInvariantTraceFormula....................................... 10
2 LocalTheory ................................................................. 25
1 CaseofDivisionAlgebras............................................... 25
2 OrbitalIntegrals.......................................................... 32
3 AutomorphicForms ..................................................... 55
4 TraceFormula............................................................ 58
5 Density ................................................................... 61
6 Characters................................................................ 66
7 Coinvariants.............................................................. 71
8 TraceFunctions.......................................................... 73
9 Stability .................................................................. 79
10 DiscreteSeries........................................................... 81
11 Decay..................................................................... 82
12 Finiteness................................................................. 84
13 SimpleAlgebras ......................................................... 85
14 Germs .................................................................... 90
15 Comparison .............................................................. 91
16 Existence................................................................. 92
17 Isolation.................................................................. 94
18 Correspondence.......................................................... 96
19 Tempered................................................................. 98
20 Irreducibility............................................................. 101
21 Unitarity.................................................................. 103
22 Induction................................................................. 105
23 CuspidalGlobalCorrespondence....................................... 107
24 ComplementsonLocalRepresentations................................ 108
ix
x Contents
25 CompleteGlobalCorrespondence...................................... 116
26 OneCuspidalPlace...................................................... 126
3 Arthur’sNoninvariantTraceFormula .................................... 139
1 PreliminaryDefinitions.................................................. 139
2 TheKernelK .x;y/...................................................... 145
P
3 AReviewofEisensteinSeries .......................................... 150
4 TheSecondFormulafortheKernel .................................... 156
5 TheModifiedKernelIdentity........................................... 163
6 SomeGeometricLemmas............................................... 166
7 IntegrabilityofkT.x;f/.................................................. 171
o
8 WeightedOrbitalIntegrals .............................................. 177
9 ATruncationOperator................................................... 183
10 IntegrabilityofkT.x;f/.................................................. 191
(cid:4)
11 TheOperatorMPT.(cid:3)/(cid:4).................................................... 201
12 EvaluationinaSpecialCase ............................................ 206
13 Conclusion............................................................... 213
4 StudyofNoninvariance ..................................................... 215
1 Notation.................................................................. 215
2 AReviewoftheTraceFormula......................................... 220
3 TheDistributionsJoandJ(cid:4) ............................................. 223
4 Noninvariance............................................................ 231
5 IntertwiningOperators................................................... 238
6 Normalization............................................................ 240
7 RealGroups.............................................................. 244
8 p-AdicGroups ........................................................... 251
9 StandardRepresentations................................................ 253
10 ConvexSetsandSomeRelatedFunctions.............................. 257
11 SomeExamples.......................................................... 265
12 TheDistributionsJM.(cid:3)(cid:5)/................................................ 273
13 TheDistributionsJM;(cid:6) andJM;(cid:2)......................................... 277
14 HeckeInvariance......................................................... 285
15 TheDistributionsJ .(cid:3);X/ .............................................. 288
M
16 Residues.................................................................. 292
17 ProofofProposition17.3................................................ 298
18 ChangesofContour...................................................... 304
19 TheSpacesH .G.F //andI .G.F // ................................ 307
ac S ac S
20 TheMap(cid:7) .............................................................. 311
M
5 TheInvariantTraceFormula............................................... 317
1 InvariantHarmonicAnalysis............................................ 321
2 TheInvariantDistributionsI .(cid:6)/....................................... 325
M
3 TheInvariantDistributionsI .(cid:3);X/.................................... 334
M
4 SomeFurtherMapsandDistributions.................................. 338
5 AContourIntegral....................................................... 347