Dirac Operators in Representation Theory

204 Pages·2006·1.191 MB·English

by  Jing-Song Huang, Pavle Pandzic

Jing-Song Huang
Pavle Pandzˇic´
Dirac Operators in
Representation Theory
Birkha¨user
Boston • Basel • Berlin
PavlePandzˇic´
Jing-SongHuang
UniversityofZagreb
TheHongKongUniversityof
DepartmentofMathematics
ScienceandTechnology
10002Zagreb
DepartmentofMathematics
Croatia
ClearWaterBay,Kowloon
HongKong,P.R.C.
MathematicsSubjectClassification(2000):22E46
LibraryofCongressControlNumber:2006923127
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Preface
We begin with the origin of the Dirac equation and give a brief introduction of the
DiracoperatorsduetoParthasarathy,VoganandKostant.Thenweexplainaconjec-
tureofVoganonDiraccohomology,whichweprovedin[HP1],itsapplicationsand
theorganizationofthebook.
0.1 The Dirac equation. The Dirac equation has an interesting connection to
E = mc2, the Einstein’s equation from his special theory of relativity. This equa-
tion relates the energy E of a particle at rest to its mass m through a conversion
factor,thesquareofthespeedc oflight.However,thiswayofwritingtheequation
obscures the underlying four-dimensional geometry. The relation for a particle in
motionisahyperbolicequation
E2−c2p·p=(mc2)2. (0.1)
Here the energy E is the first component of a vector (E,cp) = (E,cp ,cp ,cp )
1 2 3
andpisthevectorthatdescribesthemomentumoftheparticle.Thismoregeneral
equationexhibitsthemechanismoftheconversionofmassintorelativemotion.
Todescriberelativisticspin 1 particlesDiracrewrotethequadraticEinsteinrela-
2
tion(0.1)asalinearrelation.Thiswouldseemimpossible.ButDiraccameupwith
anewideabywritingtherelationasfollows:
(cid:2)3
γ E +c γ p =mc2I, (0.2)
0 j j
j=1
where γ (k = 0,1,2,3) are 4 × 4 matrices and I is the 4 × 4 identity matrix.
k
Thefourmatricesγ ,γ ,γ andγ areanticommutative:γ γ = −γ γ for j (cid:3)= k.
0 1 2 3 j k k j
Furthermore,theysatisfyγ2 = I andγ2 = −I for j (cid:3)= 0.Inquantummechanics,
0 j
energy,andmomentumareexpressedbydifferentialoperators:
∂
E =i(cid:2) , p=−i(cid:2)(cid:2),
∂t
where(cid:2)isthePlanckconstant.Substitutingtheabovedifferentialsfor E andpinto
(0.2),oneobtainstheDiracequation:
vi Preface
∂ (cid:2)3 ∂
i(cid:2)γ −i(cid:2)c γ =mc2I. (0.3)
0∂t j∂x
j=1 j
Thismatrix-valuedfirst-orderdifferentialequationhashadaremarkablesuccessin
describing many elementary particles that make up matter. The various analogs of
thecorrespondingdifferentialoperatorarecalledDiracoperators.Theimpactofthe
Diracoperatorsonthedevelopmentofmathematicsisalsosignificant.Theextension
of the definition of a Dirac operator to a differentiable manifold and a proof of the
corresponding index theorem by Atiyah and Singer are one of the most influential
theoriesofmathematicsinthetwentiethcentury.
0.2 Group representations and discrete series. Representations of finite groups
were studied by Dedekind, Frobenius, Hurwitz and Schur at the beginning of the
20thcentury.Inthe1920s,thefocusofinvestigationswasonrepresentationtheory
ofcompactLiegroupsanditsrelationstoinvarianttheory.CartanandWeylobtained
thewell-knownclassificationofequivalenceclassesofirreducibleunitaryrepresen-
tationsofconnectedcompactLiegroupsintermsofhighestweights.Inthe1930s,
Dirac and Wigner initiated the investigation of infinite-dimensional representations
ofnoncompactLiegroups.
Harish-Chandra was a Ph.D. student of Dirac at the University of Cambridge
from 1945 to 1947. After receiving his Ph.D. Harish-Chandra began a systematic
investigation of infinite-dimensional representations of semisimple Lie groups and
laid down the foundation for further development. Let G denote a semisimple Lie
groupwithfinitecenter;thediscreteseriesrepresentationsaretheirreduciblerepre-
sentationscontainedinthedecompositionoftheregularrepresentationson L2(G).
In 1965 and 1966, Harish-Chandra published two papers which gave a complete
classification of the discrete series representations. Later he used this classification
toprovethePlancherelformula.Thisclassificationofdiscreteseriesisalsocrucial
toLanglandsclassificationofadmissiblerepresentations.However,Harish-Chandra
didnotgiveanexplicitconstructionofdiscreteseries.Hisworkwasparalleltothat
ofCartan–WeylforirreducibleunitaryrepresentationsofcompactLiegroups.
0.3 Dirac cohomology and Vogan’s conjecture. The Dirac operator was used for
construction of the discrete series representations by Parthasarathy and Atiyah–
Schmid. Denote the Lie algebras of G and K by g and k , where K is a maxi-
0 0
malcompactsubgroupof G.Wedropthesubscriptfortheircomplexifications.Let
g=k⊕pbethecomplexifiedCartandecomposition.TheKillingformong,which
is nondegenerate on p, defines the Clifford algebra C(p) as an associative algebra
withunit.Givenanorthonormalbasis Z ofp,Vogandefinedanalgebraicversionof
i
theDiracoperatortobe
(cid:2)
D = Z ⊗Z ∈ U(g)⊗C(p).
i i
i
Itiseasytoseethat D isindependentofthechoiceofbasis Z and K-invariant(for
i
theadjointactionof K onbothfactors).Thenitcanbeshownthat
D2 =−(cid:4)g⊗1+(cid:4)k(cid:5) +C,
Preface vii
whereC isaconstantandk(cid:5)denotesadiagonalembeddingofkintoU(g)⊗C(p).
If S is a space of spinors (a simple C(p)-module), then D acts on X ⊗ S. The
Diraccohomologyisdefinedtobe
H (X)= Ker D/Ker D∩ Im D.
D
The following statement was conjectured by Vogan and proved in [HP1]. For any
z ∈ Z(g)thereisauniqueζ(z) ∈ Z(k(cid:5)),andthereare K-invariantelementsa,b ∈
U(g)⊗C(p),suchthat
z⊗1=ζ(z)+Da+bD.
Moreover,ζ: Z(g)→ Z(k(cid:5))isanalgebrahomomorphismhavingasimpleexplicit
descriptionintermsofHarish-Chandraisomorphisms.
This allows us to identify the infinitesimal character of an irreducible (g,K)-
modulethathasnonzeroDiraccohomology.Kostantextendsthisresulttohiscubic
DiracoperatordefinedinamoregeneralsettingofapairofquadraticLiealgebras.
0.4Applicationsandorganizationofthebook.Determinationoftheinfinitesimal
characterbyDiraccohomologyenablesustosimplifyproofsofafewclassicalthe-
orems and even sharpen some. After some preliminaries in Chapters 1 and 2, we
explainourproofofVogan’sconjectureinChapter3.InChapter4weobtainasim-
plerproofofageneralizedWeylCharacterformuladuetoGross,Kostant,Ramond
andSteinbergaswellasageneralizedBott–Borel–Weiltheorem.Chapters5and6
providethenecessarybackgroundofcohomologicalparabolicinduction.InChapter
7 we give a simpler proof of the construction and classification of the discrete se-
riesrepresentations.InChapter8wesharpentheLanglandsformulaonautomorphic
formsandobtaintherelationofDiraccohomologyto(g,K)-cohomology.Chapter
9isontherelationofDiraccohomologytoLiealgebracohomology.InChapter10
weproveananalogueofVogan’sconjectureforbasicclassicalLiesuperalgebras.
HongKong Jing-SongHuang
June,2005 PavlePandzˇic´
Contents
Preface ......................................................... v
1 LieGroups,LieAlgebrasandRepresentations.................... 1
1.1 Liegroupsandalgebras ..................................... 1
1.2 Finite-dimensionalrepresentations ............................ 9
1.3 Infinite-dimensionalrepresentations ........................... 21
1.4 Infinitesimalcharacters...................................... 26
1.5 Tensorproductsofrepresentations ............................ 30
2 CliffordAlgebrasandSpinors.................................. 33
2.1 RealCliffordalgebras....................................... 33
2.2 ComplexCliffordalgebrasandspinmodules.................... 40
2.3 SpinrepresentationsofLiegroupsandalgebras ................. 45
3 DiracOperatorsintheAlgebraicSetting......................... 57
3.1 Diracoperators ............................................ 57
3.2 DiraccohomologyandVogan’sconjecture ..................... 61
3.3 Adifferentialon(U(g)⊗C(p))K ............................. 65
3.4 Thehomomorphismζ....................................... 70
3.5 AnextensionofParthasarathy’sDiracinequality ................ 71
4 AGeneralizedBott–Borel–WeilTheorem ........................ 73
4.1 KostantcubicDiracoperators ................................ 73
4.2 Diraccohomologyoffinite-dimensionalrepresentations .......... 76
4.3 Characters ................................................ 78
4.4 AgeneralizedWeylcharacterformula ......................... 81
4.5 AgeneralizedBott–Borel–Weiltheorem ....................... 82
5 CohomologicalInduction...................................... 85
5.1 Overview ................................................. 85
5.2 Somegeneralitiesaboutadjointfunctors ....................... 89
x Contents
5.3 HomologicalalgebraofHarish-Chandramodules................ 95
5.4 Zuckermanfunctors ........................................ 101
5.5 Bernsteinfunctors .......................................... 106
6 PropertiesofCohomologicallyInducedModules .................. 115
6.1 Dualitytheorems........................................... 115
6.2 Infinitesimalcharacter, K-typesandvanishing .................. 121
6.3 Irreducibilityandunitarity ................................... 127
6.4 Aq(λ)modules............................................. 129
6.5 Unitarymoduleswithstronglyregularinfinitesimalcharacter...... 132
7 DiscreteSeries............................................... 133
7.1 L2-indextheorem .......................................... 134
7.2 Existenceofdiscreteseries................................... 137
7.3 Globalcharacters........................................... 138
7.4 Exhaustionofdiscreteseries ................................. 140
8 DimensionsofSpacesofAutomorphicForms ..................... 145
8.1 Hirzebruchproportionalityprinciple........................... 145
8.2 Dimensionsofspacesofautomorphicforms .................... 147
8.3 Diraccohomologyand(g,K)-cohomology .................... 148
8.4 Cohomologyofdiscretesubgroups............................ 150
9 DiracOperatorsandNilpotentLieAlgebraCohomology ........... 153
9.1 u-homologyandu¯-cohomologydifferentials.................... 154
9.2 Hodgedecompositioninthefinite-dimensionalcase ............. 158
9.3 Hodgedecompositionforp−-cohomologyintheunitarycase..... 160
9.4 CalculatingDiraccohomologyinstages........................ 162
9.5 Hodgedecompositionforu¯-cohomologyintheunitarycase ....... 168
9.6 HomologicalpropertiesofDiraccohomology................... 172
10 DiracCohomologyforLieSuperalgebras ........................ 177
10.1 LiesuperalgebrasofRiemanniantype ......................... 177
10.2 Diracoperatorfor(g,g ) .................................... 183
0
10.3 AnalogofVogan’sconjecture ................................ 185
10.4 DiraccohomologyforLiesuperalgebras ....................... 187
References ...................................................... 193
Index........................................................... 197
Dirac Operators in
Representation Theory