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82 Pages·2012·0.84 MB·English

Shimura varieties and moduli
J.S.Milne
Abstract. ConnectedShimuravarietiesarethequotientsofhermitiansymmetric
domainsbydiscretegroupsdefinedbycongruenceconditions.Weexaminetheir
relationwithmodulivarieties.
Contents
Notations 470
1 Ellipticmodularcurves 471
Definitionofellipticmodularcurves 471
Ellipticmodularcurvesasmodulivarieties 472
2 Hermitiansymmetricdomains 475
PreliminariesonCartaninvolutionsandpolarizations 475
Definitionofhermitiansymmetricdomains 477
Classificationintermsofrealgroups 478
Classificationintermsofrootsystems 479
Example: theSiegelupperhalfspace 480
3 DiscretesubgroupsofLiegroups 481
LatticesinLiegroups 481
Arithmeticsubgroupsofalgebraicgroups 482
ArithmeticlatticesinLiegroups 484
Congruencesubgroupsofalgebraicgroups 485
4 Locallysymmetricvarieties 486
Quotientsofhermitiansymmetricdomains 486
Thealgebraicstructureonthequotient 486
Locallysymmetricvarieties 488
Example: Siegelmodularvarieties 488
5 VariationsofHodgestructures 489
TheDelignetorus 490
RealHodgestructures 490
RationalHodgestructures 491
Polarizations 491
Localsystemsandvectorsheaveswithconnection 492
VariationsofHodgestructures 492
468 Shimuravarietiesandmoduli
6 Mumford-Tategroupsandtheirvariationinfamilies 493
Theconditions(SV) 493
DefinitionofMumford-Tategroups 494
SpecialHodgestructures 496
ThegenericMumford-Tategroup 497
VariationofMumford-Tategroupsinfamilies 498
VariationofMumford-Tategroupsinalgebraicfamilies 502
7 Periodsubdomains 502
Flagmanifolds 503
Perioddomains 503
Periodsubdomains 505
Whymodulivarietiesare(sometimes)locallysymmetric 507
Application: Riemann’stheoreminfamilies 508
8 VariationsofHodgestructuresonlocallysymmetricvarieties 509
ExistenceofHodgestructuresofCM-typeinafamily 509
DescriptionofthevariationsofHodgestructuresonD(Γ) 510
ExistenceofvariationsofHodgestructures 512
9 AbsoluteHodgeclassesandmotives 514
Thestandardcohomologytheories 514
AbsoluteHodgeclasses 515
ProofofDeligne’stheorem 518
MotivesforabsoluteHodgeclasses 520
10 SymplecticRepresentations 524
Preliminaries 524
Therealcase 524
Therationalcase 531
11 Moduli 536
Mumford-Tategroups 536
Familiesofabelianvarietiesandmotives 539
Shimuravarieties 540
Shimuravarietiesasmodulivarieties 541
References 544
Index 548
Introduction
Thehermitiansymmetricdomainsarethecomplexmanifoldsisomorphicto
boundedsymmetricdomains.TheGriffithsperioddomainsaretheparameterspaces
forpolarizedrationalHodgestructures. Aperioddomainisahermitiansymmetric
domain if the universal family of Hodge structures on it is a variation of Hodge
J.S.Milne 469
structures,i.e.,satisfiesGriffithstransversality. Thisrarelyhappens,but,asDeligne
showed,everyhermitiansymmetricdomaincanberealizedasthesubdomainofa
perioddomainonwhichcertaintensorsfortheuniversalfamilyareoftype(p,p)
(i.e.,areHodgetensors).
Inparticular,everyhermitiansymmetricdomaincanberealizedasamoduli
spaceforHodgestructuresplustensors. Thisalltakesplaceintheanalyticrealm,
because hermitian symmetric domains are not algebraic varieties. To obtain an
algebraicvariety,wemustpasstothequotientbyanarithmeticgroup. Infact,in
order to obtain a moduli variety, we should assume that the arithmetic group is
definedbycongruenceconditions. Thealgebraicvarietiesobtainedinthiswayare
theconnectedShimuravarieties.
The arithmetic subgroup lives in a semisimple algebraic group over Q, and
thevariationsofHodgestructuresontheconnectedShimuravarietyareclassified
intermsofauxiliaryreductivealgebraicgroups. Inordertorealizetheconnected
Shimuravarietyasamodulivariety,wemustchoosetheadditionaldatasothatthe
variationofHodgestructuresisofgeometricorigin.
The main result of the article classifies the connected Shimura varieties for
which this is known to be possible. Briefly, in a small number of cases, the con-
nectedShimuravarietyisamodulivarietyforabelianvarietieswithpolarization,
endomorphism,andlevelstructure(thePELcase);foramuchlargerclass,thevariety
isamodulivarietyforabelianvarietieswithpolarization, Hodgeclass, andlevel
structure(thePHLcase);forallconnectedShimuravarietiesexceptthoseoftype
E ,E ,andcertaintypesD,thevarietyisamodulivarietyforabelianmotiveswith
6 7
additionalstructure. Intheremainingcases,theconnectedShimuravarietyisnot
amodulivarietyforabelianmotives, anditisnotknownwhetheritisamoduli
varietyatall.
Wenowsummarizethecontentsofthearticle.
§1.Asanintroductiontothegeneraltheory,wereviewthecaseofellipticmod-
ularcurves. Inparticular,weprovethatthemodularcurveconstructedanalytically
coincideswiththemodularcurveconstructedalgebraicallyusinggeometricinvariant
theory.
§2. Webrieflyreviewthetheoryofhermitiansymmetricdomains. Togivea
hermitiansymmetricdomainamountstogivingarealsemisimpleLiegroupHwith
trivialcentreandahomomorphismufromthecirclegrouptoHsatisfyingcertain
conditions. Thisleadstoaclassificationofhermitiansymmetricdomainsinterms
ofDynkindiagramsandspecialnodes.
§3. The group of holomorphic automorphisms of a hermitian symmetric
domain is a real Lie group, and we are interested in quotients of the domain by
certaindiscretesubgroupsofthisLiegroup.Inthissectionwereviewthefundamental
theorems of Borel, Harish-Chandra, Margulis, Mostow, Selberg, Tamagawa, and
othersconcerningdiscretesubgroupsofLiegroups.
470 Shimuravarietiesandmoduli
§4. Thearithmeticlocallysymmetricvarieties(resp. connectedShimuravari-
eties)arethequotientsofhermitiansymmetricdomainsbyarithmetic(resp.con-
gruence)groups. WeexplainthefundamentaltheoremsofBailyandBorelonthe
algebraicityofthesevarietiesandofthemapsintothem.
§5. WereviewthedefinitionofHodgestructuresandoftheirvariations,and
statethefundamentaltheoremofGriffithsthatmotivatedtheirdefinition.
§6. WedefinetheMumford-TategroupofarationalHodgestructure,andwe
provethebasicresultsconcerningtheirbehaviourinfamilies.
§7. Wereviewthetheoryofperioddomains,andexplainDeligne’sinterpreta-
tionofhermitiansymmetricdomainsasperiodsubdomains.
§8. We classify certain variations of Hodge structures on locally symmetric
varietiesintermsofgroup-theoreticdata.
§9.Inordertobeabletorealizeallbutahandfuloflocallysymmetricvarieties
asmodulivarieties,weshallneedtoreplacealgebraicvarietiesandalgebraicclasses
bymoregeneralobjects. Inthissection,weproveDeligne’stheoremthatallHodge
classesonabelianvarietiesareabsolutelyHodge,andhavealgebraicmeaning,and
wedefineabelianmotives.
§10. Following Satake and Deligne, we classify the symplectic embeddings
ofanalgebraicgroupthatgiverisetoanembeddingoftheassociatedhermitian
symmetricdomainintoaSiegelupperhalfspace.
§11. WeusetheresultsoftheprecedingsectionstodeterminewhichShimura
varietiescanberealizedasmodulivarietiesforabelianvarieties(orabelianmotives)
plusadditionalstructure.
Althoughtheexpertwillfindlittlethatisnewinthisarticle,thereismuchthat
isnotwellexplainedintheliterature. Asfaraspossible,completeproofshavebeen
included.
Notations
We use k to denote the base field (always of characteristic zero), and kal to
denoteanalgebraicclosureofk. “Algebraicgroup”means“affinealgebraicgroup
scheme”and“algebraicvariety”means“geometricallyreducedschemeoffinitetype
overafield”. ForasmoothalgebraicvarietyXoverC,weletXan denotethesetX(C)
endowedwithitsnaturalstructureofacomplexmanifold. Thetangentspaceata
pointpofspaceXisdenotedbyT (X).
p
Vectorspacesandrepresentationsarefinitedimensionalunlessindicatedoth-
erwise. ThelineardualofavectorspaceV isdenotedbyV∨. Forak-vectorspace
V andcommutativek-algebraR,V =R⊗ V. ForatopologicalspaceS,weletV
R k S
denotetheconstantlocalsystemofvectorspacesonSdefinedbyV. Byalatticeina
realvectorspace,wemeanafulllattice,i.e.,theZ-modulegeneratedbyabasisfor
thevectorspace.
J.S.Milne 471
A vector sheafon a complex manifold (or scheme) S is a locally free sheaf
of O -modules of finite rank. In order for W to be a vector subsheaf of a vector
S
sheafV,werequirethatthemapsonthefibresW → V beinjective. Withthese
s s
definitions,vectorsheavescorrespondtovectorbundlesandvectorsubsheavesto
vectorsubbundles.
ThequotientofaLiegrouporalgebraicgroupGbyitscentreZ(G)isdenoted
byGad. ALiegrouporalgebraicgroupissaidtobeadjoint ifitissemisimple(in
particular,connected)withtrivialcentre. Analgebraicgroupissimple(resp. almost
simple)ifitconnectednoncommutativeandeverypropernormalsubgroupistrivial
(resp. finite). Anisogeny ofalgebraicgroupsisasurjectivehomomorphismwith
finitekernel. AnalgebraicgroupGissimplyconnectedifitissemisimpleandevery
isogenyG(cid:48) →GwithG(cid:48) connectedisanisomorphism. Theinnerautomorphismof
Gdefinedbyanelementgisdenotedbyinn(g). Letad: G→Gad bethequotient
map. ThereisanactionofGad onGsuchthatad(g)actsasinn(g)forallg∈G(kal).
ForanalgebraicgroupGoverR,G(R)+ istheidentitycomponentofG(R)forthe
realtopology. ForafiniteextensionoffieldsL/kandanalgebraicgroupGoverL,
wewrite(G) foralgebraicgroupoverkobtainedby(Weil)restrictionofscalars.
L/k
Asusual,G =GL andµ isthekernelofG −N→G .
m 1 N m m
AprimeofanumberfieldkisaprimeidealinO (afiniteprime),anembedding
k
ofkintoR(arealprime),oraconjugatepairofembeddingsofkintoC(acomplex
(cid:16)(cid:81) (cid:17)
prime). TheringoffiniteadèlesofQisA =Q⊗ Z .
f p p
Weuseιorz(cid:55)→z¯ todenotecomplexconjugationonCoronasubfieldofC,
andweuseX(cid:39)Y tomeanthatXandY isomorphicwithaspecificisomorphism—
whichisomorphismshouldalwaysbeclearfromthecontext.
Foralgebraicgroupsweusethelanguageofmodernalgebraicgeometry,not
themoreusuallanguage,whichisbasedonWeil’sFoundations. Forexample,ifG
andG(cid:48) arealgebraicgroupsoverafieldk,thenbyahomomorphismG → G(cid:48) we
meanahomomorphismdefinedoverk,notoversomeuniversaldomain. Similarly,
asimplealgebraicgroupoverafieldkneednotbegeometrically(i.e.,absolutely)
simple.
1. Elliptic modular curves
The first Shimura varieties, and the first moduli varieties, were the elliptic
modularcurves. Inthissection,wereviewthetheoryofellipticmodularcurvesas
anintroductiontothegeneraltheory.
Definitionofellipticmodularcurves
LetDbethecomplexupperhalfplane,
D={z∈C|(cid:61)(z)>0}.
472 Shimuravarietiesandmoduli
ThegroupSL (R)actstransitivelyonDbytherule
2
(cid:32) (cid:33)
ab az+b
z= .
c d cz+d
A subgroup Γ of SL (Z) is a congruence subgroup if, for some integer N (cid:62) 1, Γ
2
containstheprincipalcongruencesubgroupoflevelN,
Γ(N)d=ef{A∈SL (Z)|A≡ImodN}.
2
AnellipticmodularcurveisthequotientΓ\DofDbyacongruencegroupΓ.Initially
thisisaone-dimensionalcomplexmanifold,butitcanbecompactifiedbyadding
afinitenumberof“cusps”,andsoithasauniquestructureofanalgebraiccurve
compatiblewithitsstructureasacomplexmanifold.1 Thiscurvecanberealized
asamodulivarietyforellipticcurveswithlevelstructure,fromwhichitispossible
deducemanybeautifulpropertiesofthecurve,forexample,thatithasacanonical
modeloveraspecificnumberfield,andthatthecoordinatesofthespecialpointson
themodelgenerateclassfields.
Ellipticmodularcurvesasmodulivarieties
ForanellipticcurveEoverC,theexponentialmapdefinesanexactsequence
(1.1) 0→Λ→T (Ean)−ex→p Ean →0
0
with
Λ(cid:39)π (Ean,0)(cid:39)H (Ean,Z).
1 1
ThefunctorE (cid:32) (T E,Λ)isanequivalencefromthecategoryofcomplexelliptic
0
curvestothecategoryofpairsconsistingofaone-dimensionalC-vectorspaceanda
lattice. Thus,togiveanellipticcurveoverCamountstogivingatwo-dimensional
R-vectorspaceV,acomplexstructureonV,andalatticeinV. ItisknownthatD
parametrizesellipticcurvesplusadditionaldata.Traditionally,toapointτofDone
attachesthequotientofCbythelatticespannedby1andτ. Inotherwords,one
fixestherealvectorspaceandthecomplexstructure,andvariesthelattice. Fromthe
pointofviewofperioddomainsandShimuravarieties,itismorenaturaltofixthe
realvectorspaceandthelattice,andvarythecomplexstructure.2
Thus,letV beatwo-dimensionalvectorspaceoverR.AcomplexstructureonV
isanendomorphismJofVsuchthatJ2 =−1.FromsuchaJ,wegetadecomposition
VC = VJ+ ⊕VJ− of VC into its +i and −i eigenspaces, and the isomorphism V →
VC/VJ− carries the complex structure J on V to the natural complex structure on
VC/VJ−. ThemapJ(cid:55)→VC/VJ− identifiesthesetofcomplexstructuresonV withthe
1WeareusingthatthefunctorS(cid:32)SanfromsmoothalgebraicvarietiesoverCtocomplexmanifolds
definesanequivalencefromthecategoryofcompletesmoothalgebraiccurvestothatofcompactRiemann
surfaces.
2ThechoiceofatrivializationofavariationofintegralHodgestructuresattachestoeachpointofthe
underlyingspaceafixedrealvectorspaceandlattice,butavaryingHodgestructure—seebelow.
J.S.Milne 473
setofnonrealone-dimensionalquotientsofVC,i.e.,withP(VC)(cid:114)P(V). Thisspace
hastwoconnectedcomponents.
NowchooseabasisforV,andidentifyitwithR2. Letψ: V ×V → Rbethe
alternatingform
ψ((a),(c))=det(a c)=ad−bc.
b d bd
Ononeoftheconnectedcomponents,whichwedenoteD,thesymmetricbilinear
form
(x,y)(cid:55)→ψ (x,y)d=efψ(x,Jy): V×V →R
J
is positive definite and on the other it is negative definite. Thus D is the set of
complexstructuresonV forwhich+ψ(ratherthan−ψ)isaRiemannform. Our
choiceofabasisforV identifiesP(VC)(cid:114)P(V)withP1(C)(cid:114)P1(R)andDwiththe
complexupperhalfplane.
NowletΛbethelatticeZ2 inV. ForeachJ ∈ D,thequotient(V,J)/Λisan
ellipticcurveEwithH (Ean,Z)(cid:39)Λ. Inthisway,weobtainaone-to-onecorrespon-
1
dencebetweenthepointsofDandtheisomorphismclassesofpairsconsistingof
anellipticcurveEoverCandanorderedbasisforH (Ean,Z).
1
LetE denotethekernelofmultiplicationbyNonanellipticcurveE. Thus,
N
forthecurveE=(V,J)/Λ,
E (C)= 1Λ/Λ(cid:39)Λ/NΛ≈(Z/NZ)2.
N N
Alevel-NstructureonEisapairofpointsη=(t ,t )inE(C)thatformsanordered
1 2
basisforE (C).
N
ForanellipticcurveEoveranyfield,thereisanalgebraicallydefined(Weil)
pairing
e : E ×E →µ .
N N N N
WhenthegroundfieldisC,thisinducesanisomorphism(cid:86)2(E (C))(cid:39)µ (C). In
N N
thefollowing,wefixaprimitiveNthrootζof1inC,andwerequirethatourlevel-N
structuressatisfytheconditione (t ,t )=ζ.
N 1 2
IdentifyΓ(N)withthesubgroupofSL(V)whoseelementspreserveΛandactas
theidentityonΛ/NΛ. OnpassingtothequotientbyΓ(N),weobtainaone-to-one
correspondencebetweenthepointsofΓ(N)\Dandtheisomorphismclassesofpairs
consistingofanellipticcurveEoverCandalevel-NstructureηonE.LetY denote
N
thealgebraiccurveoverCwithYan =Γ(N)\D.
N
Letf: E→SbeafamilyofellipticcurvesoveraschemeS,i.e.,aflatmapof
schemestogetherwithasectionwhosefibresareellipticcurves. Alevel-Nstructure
onE/Sisanorderedpairofsectionstofthatgivealevel-NstructureonE foreach
s
closedpointsofS.
Proposition1.2. Letf: E→Sbeafamilyofellipticcurvesonasmoothalgebraiccurve
SoverC, andletηbealevel-NstructureonE/S. Themapγ: S(C) → Y (C)sending
N
474 Shimuravarietiesandmoduli
s ∈ S(C) to the point of Γ(N)\D corresponding to (E ,η ) is regular, i.e., defined by a
s s
morphismofalgebraiccurves.
Proof. Wefirstshowthatγisholomorphic. Forthis,weusethatP(VC)istheGrass-
mannmanifoldclassifyingtheone-dimensionalquotientsofVC. Thismeansthat,
foranycomplexmanifoldMandsurjectivehomomorphismα: OM⊗RV →Wof
vector sheaves on M with W of rank 1, the map sending m ∈ M to the point of
P(VC)correspondingtothequotientαm: VC →Wm ofVC isholomorphic.
Letf: E → Sbeafamilyofellipticcurvesonaconnectedsmoothalgebraic
varietyS. TheexponentialmapdefinesanexactsequenceofsheavesonSan
0−→R f Z−→T (Ean/San)−→Ean −→0
1 ∗ 0
whosefibreatapoints∈San isthesequence(1.1)forE . Fromthefirstmapinthe
s
sequencewegetasurjectivemap
(1.3) OSan ⊗ZR1f∗Z(cid:16)T0(Ean/San).
Let(t ,t )bealevel-NstructureonE/S. EachpointofSan hasanopenneighbour-
1 2
hoodUsuchthatt | andt | lifttosectionst˜ andt˜ ofT (Ean/San)overU;now
1U 2U 1 2 0
Nt˜ andNt˜ aresectionsofR f ZoverU,andtheydefineanisomorphism
1 2 1 ∗
Z2 →R f Z| .
U 1 ∗ U
OntensoringthiswithO ,
Uan
OUan ⊗ZZ2U →OUan ⊗R1f∗Z|U
andcomposingwith(1.3),wegetasurjectivemap
OUan ⊗RV (cid:16)T0(Ean/San)|U
ofvectorsheavesonU,whichdefinesaholomorphicmapU→P(VC). Thismaps
intoD,anditscompositewiththequotientmapD→Γ(N)\Disthemapγ. There-
foreγisholomorphic.
Itremainstoshowthatγisalgebraic. WenowassumethatSisacurve. After
passingtoafinitecovering,wemaysuSpposethatNiseven. LetY¯ (resp. S¯)be
N
thecompletionofY (resp. S)toasmoothcompletealgebraiccurve. Wehavea
N
holomorphicmap
San −γ→Yan ⊂Y¯an;
N N
toshowthatitisregular,itsufficestoshowthatitextendstoaholomorphicmapof
compactRiemannsurfacesS¯an →Y¯an. ThecurveY isisomorphictoP1(cid:114){0,1,∞}.
N 2
Thecomposedmap
San −γ→Yan −on→to Yan ≈P1(C)(cid:114){0,1,∞}
N 2
J.S.Milne 475
doesnothaveanessentialsingularityatanyofthe(finitelymany)pointsofS¯an(cid:114)
San becausethiswouldviolatethebigPicardtheorem.3 Therefore,itextendstoa
holomorphicmap S¯an → P1(C), whichimplies thatγextendsto aholomorphic
mapγ¯: S¯an →Y¯an,asrequired. (cid:3)
N
Let F be the functor sending a scheme S of finite type over C to the set of
isomorphismclassesofpairsconsistingofafamilyellipticcurvesf: E→SoverS
andalevel-NstructureηonE. AccordingtoMumford[44],Chapter7,thefunctor
FisrepresentablewhenN(cid:62)3. Moreprecisely,whenN(cid:62)3thereexistsasmooth
algebraiccurveS overCandafamilyofellipticcurvesoverS endowedwitha
N N
levelNstructurethatisuniversalinthesensethatanysimilarpaironaschemeSis
isomorphictothepullbackoftheuniversalpairbyauniquemorphismα: S→S .
N
Theorem1.4. Thereisacanonicalisomorphismγ: S →Y .
N N
Proof. AccordingtoProposition1.2,theuniversalfamilyofellipticcurveswithlevel-
N structure on S defines a morphism of smooth algebraic curves γ: S → Y .
N N N
Both sets S (C) and Y (C) are in natural one-to-one correspondence with the
N N
set of isomorphism classes of complex elliptic curves with level-N structure, and
γ sends the point in S (C) corresponding to a pair (E,η) to the point in Y (C)
N N
correspondingtothesamepair. Therefore,γ(C)isbijective,whichimpliesthatγis
anisomorphism. (cid:3)
Inparticular,wehaveshownthatthecurveS ,constructedbyMumfordpurely
N
intermsofalgebraicgeometry,isisomorphicbytheobviousmaptothecurveY ,
N
constructedanalytically. Ofcourse,thisiswellknown,butitisdifficulttofinda
proof of it in the literature. For example, Brian Conrad has noted that it is used
withoutreferencein[30].
Theorem1.4saysthatthereexistsasinglealgebraiccurveoverCenjoyingthe
goodpropertiesofbothS andY .
N N
2. Hermitian symmetric domains
The natural generalization of the complex upper half plane is a hermitian
symmetricdomain.
PreliminariesonCartaninvolutionsandpolarizations
LetGbeaconnectedalgebraicgroupoverR,andletσ : g(cid:55)→g¯ denotecomplex
0
conjugationonGC withrespecttoG. ACartaninvolutionofGisaninvolutionθof
G(asanalgebraicgroupoverR)suchthatthegroup
G(θ)(R)={g∈G(C)|g=θ(g¯)}
3Recallthatthissaysthataholomorphicfunctiononthepunctureddiskwithanessentialsingularity
at0omitsatmostonevalueinC.Thereforeafunctiononthepunctureddiskthatomitstwovalueshas
(atworst)apoleat0,andsoextendstoafunctionfromthewholedisktoP1(C).
476 Shimuravarietiesandmoduli
iscompact.ThenG(θ)isacompactrealformofGC,andθactsonG(C)asσ0σ=σσ0
whereσdenotescomplexconjugationonGC withrespecttoG(θ).
Consider,forexample,thealgebraicgroupGL attachedtoarealvectorspace
V
V. ThechoiceofabasisforV determinesatransposeoperatorg(cid:55)→gt,andθ: g(cid:55)→
(gt)−1 isaCartaninvolutionof GL becauseGL(θ)(R) istheunitarygroup. The
V V
basisdeterminesanisomorphismGL (cid:39)GL ,andσ (A)=A¯ andσ(A)=(A¯t)−1
V n 0
forA∈GL (C).
n
AconnectedalgebraicgroupGhasaCartaninvolutionifandonlyifithasa
compactrealform,whichisthecaseifandonlyifGisreductive. AnytwoCartan
involutions of G are conjugate by an element of G(R). In particular, all Cartan
involutionsofGL arise,asinthelastparagraph,fromthechoiceofabasisforV.
V
AnalgebraicsubgroupGofGL isreductiveifandonlyifitisstableunderg(cid:55)→gt
V
for some basis of V, in which case the restriction of g (cid:55)→ (gt)−1 to G is a Cartan
involution. EveryCartaninvolutionofGisofthisform. See[53],I,§4.
LetCbeanelementofG(R)whosesquareiscentral(soinn(C)isaninvolution).
AC-polarizationonarealrepresentationV ofGisaG-invariantbilinearformϕ: V×
V →Rsuchthattheformϕ : (x,y)(cid:55)→ϕ(x,Cy)issymmetricandpositivedefinite.
C
Theorem2.1. Ifinn(C)isaCartaninvolutionofG,theneveryfinitedimensionalreal
representationofGcarriesaC-polarization;conversely,ifonefaithfulfinitedimensional
realrepresentationofGcarriesaC-polarization,theninn(C)isaCartaninvolution.
Proof. An R-bilinear form ϕ on a real vector space V defines a sesquilinear form
ϕ(cid:48): (u,v) (cid:55)→ ϕC(u,v¯)onV(C), andϕ(cid:48) ishermitian(andpositivedefinite)ifand
onlyifϕissymmetric(andpositivedefinite).
LetG→GL bearepresentationofG. Ifinn(C)isaCartaninvolutionofG,
V
thenG(innC)(R)iscompact,andsothereexistsaG(innC)-invariantpositivedefinite
symmetricbilinearformϕonV. ThenϕC isG(C)-invariant,andso
ϕ(cid:48)(gu,(σg)v)=ϕ(cid:48)(u,v), forallg∈G(C),u,v∈VC,
where σ is the complex conjugation on GC with respect to G(innC). Now σg =
inn(C)(g¯)=inn(C−1)(g¯),andso,onreplacingvwithC−1vintheequality,wefind
that
ϕ(cid:48)(gu,(C−1g¯C)C−1v)=ϕ(cid:48)(u,C−1v), forallg∈G(C),u,v∈VC.
Inparticular,ϕ(gu,C−1gv)=ϕ(u,C−1v)wheng∈G(R)andu,v∈V. Therefore,
ϕ isG-invariant. As(ϕ ) =ϕ,weseethatϕisaC-polarization.
C−1 C−1 C
Fortheconverse,oneshowsthat,ifϕisaC-polarizationonafaithfulrepresen-
tation,thenϕ isinvariantunderG(innC)(R),whichisthereforecompact. (cid:3)
C
2.2. Variant. LetGbeanalgebraicgroupoverQ,andletCbeanelementofG(R)
whosesquareiscentral.AC-polarizationonaQ-representationVofGisaG-invariant
bilinearformϕ: V ×V → QsuchthatϕR isaC-polarizationonVR. Inorderto