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Automorphicrepresentationsandcohomologyofautomorphicvectorbundles

totallyrealMaincontext SpecialcaseofGb or Pes Gh Generalcomments

Importantcomments

Let G Gk be a reductivegroupover QLet Z Gm denotethecenterofGFix aunitary character w zf FCA s e E

Thismeans YA Ex IRIo sF eatis forsome sCIR

Fix Ko IR SOG amailcompactsubgroupofG RA G w fsmoothfunctions 4 GC NG I E

s t i g z x w z CfCA Hz f 2 A axe GCA

z 7 anopencompactsubgroupK E GAgs t 9 xu GG fue K His9g ggk

IG g is ko finite i e thesubspacegeneratedby Ks g isfindim l4 For 2 g centerofuniversalenvelopingalgebraofof

Casimiroperatorcentralderivation

E Ft FE tIHg is 2 g finite i e thesubspacegenby 2 g g isfindinl

5 GrowthconditionatcuspsWill pretendthat GW YEAR iscompacttoday so nocusp

Thiswillexclude G Ghz butwe pretendthisisokay willbemissingEisensteinseries

Then A G w S GGAIl m

automorphicmultiplicity

O J I l l

Tho OgTle spectraldecompositionUR GYQe

isthe restrictedtensorproducti e foreach allbutfinitelymany l te is an unramifiedprincipalserieslet Vf sphericalvectorin tie

ThenXp't is spannedbytensorsOxeve wherewetoeforallbutfinitelymany lHere unramifiedprincipalseries Te IndB Xe

for aBorelsubgpB withquotientT D EE F EEXi Tl Tl 2e Ex a character

xQe 2 2 exx y s Tekpvely

As G Qe D Qe Gte Iwasawadecomposition

so i g e Ind.fiejdXeGIe

g g G bk Xe b gCD forgbhfBQeGtgD Tied e E alsoneed X tobetrivial on

gas BC e nG2p B Ipsphericalvector Vi a 1

Definition Anautomorphicrepresentation is an reph IT The te ofGCA thatappearin A G w forsome w

Fix an algebraicrep'nWof Kse.g Xk Ks lR SOG G bypreviouslecturethiscorresponds

r z z towk

Fix an opencompactsubgroupkg E G Ag

WygWGlAtYkfx RtFWM0 vectorbundleassoc.toW

rxshtgfkf g.pl GHtYKfxGRYka o justalocallysymmetnspacerealmanifold

ccontfshglkf.tt fE sectionsofwffgiGiQXGCA sW

ftfG wkt Wcuspissue

CaseofGk Cont Sho Weh A Gw Wek 1 dimegksymH Wh

TL

o Xk

Hot Shock.HN yQGykfjomtDxo KgXkJFQTcomesfrom lowestweightvectors into

seeegthis.chImth

peI.s IordaIiitffo Cconfusinga.w.su

GoldfeldHundley87 Ths The The They k oofactionlooksKoactsviaXk he Im like aVermamodule

aantiwithintegerweights

G antiholomorphicdiscreteseries

Tls I K y IL z If he Ou v U SgcutionlooksX k 4X.kz X k

w like a VermamoduleE E

3 principalseries To The0 0 2 infinitebothways

I p f IYyationlookslikeVermamodule

withnonintegerweights

4 finitedimereph doesn'tappearhere

5 limitofdiscreteseries wtf forms

Explain oldlnewformtheory in GbConsideronlythoseautomorphicnephsawhoseTio is a discreteseriesofwth

Split Hot shot GET aE e

alwaysone dinil ignoreconsider F TCN ptN is K F K Gkl2p

UlF TINnToCp K4Iwp Iwf474 45

Then Sµ r ftp.khjom 1264

t.fi I ttiiIE.irSalt g Iwp

Newforms If Em tip specialforGbtheorythose it sit Tip but tiptop to forthesetp 1,5741din

Ilford If Tp is an unramifiedprincipalseriesqyGht4h v fq5TwP is amisom

7pmunlessTpeigenvalueisstrange

2dim'd bc GLdQp13 Iwp 2

Ingeneral BGQpkt YIwp Weylgp

Now move tothegeneralsituation

Recall Cont Shockey hi ft'tm

Ths Wk

Let's assumethat we are inthesituationofShimuravarieties

Wewant cohomologyforholomorphicsectionsfH ShoKf We

Useresolution o Oshq E EE e

dims

o

fee I NETbc we'velearned Tangent 9 d E

tensorwith hi resolutionof who

tf shocks e H'tf fight f Hom nCPT w

0 toH'tft Ks to wnm

CfKb cohomologyWhat is Cf Ko cohomology Eg Borel Wallach ChapDLiealgebracohomology

g Liealgebra G V vectorspaceDefine C9 g v Hom notg v

d C9 g v Cot g V isgivenbydflxo xq I CDiXi f xo Ii xq tEgCDi f lxixj xo I xj xg

Thecohomology is H g v with Ho g v VotoRelative lie algebracohomologyk eg Lie subalgebra G V vectorspaceDefine d g k v Homer H 91k V e9 g v

Il

fiof V s f f x g dependsonly on each x c91k

Ff x x xD xp x fCx xpforxEkCanshowthatd sendsCHgh V into d g h VThecohomology is H g k V

g ggsk cohomologyand gsk cohomology

Letg be a Liealg notnecessarily reductive soeitherg or gK max'dcompactsubgroup K Ks 3 KE connectedcomponentof KAssumethat K is reductive

Define it g K V Homkfittgh V c9fg.fi.DKKo

So we have H g k v H g k D

theorem H shaky a 49 H 9 Kaito W

DeepTheoremWhen its is a discreteseriesor limitofdiscreteseries WiredH 9 Kcs To W is nonzeroatexactlyonedegfm.eeifGCR isconnected

inthiscase dim 1Example G Ghz To discreteseries th The01

Hoff Kes W_f t din't

H 9 Kis Fhs Wh2 I din lmy wt2

toTIE.IEtyh.zlOTTkIXkz pwt 2

Example F totallyrealfield G Rest PGLweight b he Home heall evenno i fTEHomFIR heEof

H shaky wht tf H 9e Kae XkeT I TEHomBR

multiplicityoneholdsforPGL

Concentrated in onedegree dintindegree0 ifhe 2indegree 1 ifkesoh

gree SkSo H shocks wht is concentrated in degreemo

when V is an algebraic Q rep'nof G def'd over a numberfieldis I locallyconstantsheafassoe.toV

1Shaky U I Osho deRhamlocalsystem

1 TshockfD

Get Hffshocky e I lH Shetty U U Nshd 5 U rs

Shettyco resolution

If H't Hom aCpt Ncpt V

tightm

H fg Ko To VExample Ftotallyreal G ResetGLF

Weight ht Home w he wmod2 he 2A CR't E a m

H

t th Osho Hid'YA ShdkfD HidSholkstle

Get HH Shim'the 4th

dikgs7htxofitH.ijoEkH

fshlkpHtkwY H fshlk Htt r c

Itsefsholkft 71 It shdkf 71dB Rsha cuspcusp

If Tiff H glz.IRSOkl ThsxO yhniLautom.aephofwtke

cuspL2dins'd

o Hoff Ks Tcs Xm H gkKo The H q KsThoKh o

Langlandsobservation dimHmid g k VD dim repofGofhighestwtf

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