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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
0
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