| I.AlgebraicGeometry 0.SomeCommutativeAlgebra 1.AffineandProjectiveVarieties 1.1 IdealsandAflineVarieties 1.2 ZariskiTopologyonAffineSpace 1.3 IrreducibleComponents 1.4 ProductsofAffineVarieties 1.5 AffineAlgebrasandMorphisms 1.6 ProjectiveVarieties 1.7 ProductsofProjectiveVarieties 1.8 FlagVarieties 2.Varieties 2.1 LocalRings 2.2 Prevarieties 2.3 Morphisms 2.4 Products 2.5 HausdorffAxiom 3.Dimension 3.1 DimensionofaVariety 3.2 DimensionofaSubvariety 3.3 DimensionTheorem 3.4 Consequences 4.Morphisms 4.1 FibresofaMorphism 4.2 FiniteMorphisms 4.3 ImageofaMorphism 4.4 ConstructibleSets 4.5 OpenMorphisms 4.6 BijectiveMorphisms 4.7 BirationalMorphisms 5.TangentSpaces 5.1 ZariskiTangentSpace 5.2 ExistenceofSimplePoints 5.3 LocalRingofaSimplePoint 5.4 DifferentialofaMorphism 5.5 DifferentialCriterionforSeparability 6.CompleteVarieties 6.1 BasicProperties 6.2 CompletenessofProjectiveVarieties 6.3 VarietiesIsomorphictoP 6.4 AutomorphismsofP II.AflineAlgebraicGroups 7.BasicConceptsandExamples 7.1 TheNotionofAlgebraicGroup 7.2 SomeClassicalGroups 7.3 IdentityComponent 7.4 SubgroupsandHomomorphisms 7.5 GenerationbyIrreducibleSubsets 7.6 HopfAIgebras 8.ActionsofAlgebraicGroupsonVarieties 8.1 GroupActions 8.2 ActionsofAlgebraicGroups 8.3 ClosedOrbits 8.4 SemidirectProducts 8.5 TranslationofFunctions 8.6 LinearizationofAffineGroups III.LieAlgebras 9.LieAlgebraofanAlgebraicGroup 9.1 LieAlgebrasandTangentSpaces 9.2 Convolution 9.3 Examples 9.4 SubgroupsandLieSubalgebras 9.5 DualNumbers 10.Differentiation 10.1 SomeElementaryFormulas 10.2 DifferentialofRightTranslation 10.3 TheAdjointRepresentation 10.4 DifferentialofAd 10.5 Commutators 10.6 Centralizers 10.7 AutomorphismsandDerivations IV.HomogeneousSpaces 11.ConstructionofCertainRepresentations 11.1 ActiononExteriorPowers 11.2 ATheoremofChevalley 11.3 PassagetoProjectiveSpace 11.4 CharactersandSemi-lnvariants 11.5 NormalSubgroups 12.Quotients 12.1 UniversalMappingProperty 12.2 TopologyofY 12.3 FunctionsonY 12.4 Complements 12.5 Characteristic0 V.Characteristic0Theory 13.CorrespondenceBetweenGroupsandLieAlgebras 13.1 TheLatticeCorrespondence 13.2 InvariantsandInvariantSubspaces 13.3 NormalSubgroupsandIdeals 13.4 CentersandCentralizers 13.5 SemisimpleGroupsandLieAlgebras 14.SemisimpleGroups 14.1 TheAdjointRepresentation 14.2 SubgroupsoraSemisimpleGroup 14.3 CompleteReducibilityofRepresentations VI.SemisimpleandUnipotentElements 15.Jordan-ChevalleyDecomposition 15.1 DecompositionofaSingleEndomorphism 15.2 GL（n，K）andgl（n，K） 15.3 JordanDecompositioninAlgebraicGroups 15.4 CommutingSetsofEndomorphisms 15.5 StructureofCommutativeAlgebraicGroups 16.DiagonalizableGroups 16.1 Charactersandd-Groups 16.2 Tori 16.3 RigidityofDiagonalizableGroups 16.4 WeightsandRoots VII.SolvableGroups 17.NilpotentandSolvableGroups 17.1 AGroup-TheoreticLemma 17.2 CommutatorGroups 17.3 SolvableGroups 17.4 NilpotentGroups 17.5 UnipotentGroups 17.6 Lie-KolchinTheorem 18.SemisimpleElements 18.1 GlobalandInfinitesimalCentralizers 18.2 ClosedConjugacyClasses 18.3 ActionofaSemisimpleElementonaUnipotentGroup 18.4 ActionofaDiagonalizableGroup 19.ConnectedSolvableGroups 19.1 AnExactSequence 19.2 TheNilpotentCase 19.3 TheGeneralCase 19.4 NormalizerandCentralizer 19.5 SolvableandUnipotentRadicals 20.OneDimensionalGroups 20.1 CommutativityofG 20.2 VectorGroupsande-Groups 20.3 Propertiesofp-Polynomials 20.4 AutomorphismsofVectorGroups 20.5 TheMainTheorem VIII.BorelSubgroups 21.FixedPointandConjugacyTheorems 21.1 ReviewofCompleteVarieties 21.2 FixedPointTheorem 21.3 ConjugacyofBorelSubgroupsandMaximalTori 21.4 FurtherConsequences 22.DensityandConnectednessTheorems 22.1 TheMainLemma 22.2 DensityTheorem 22.3 ConnectednessTheorem 22.4 BorelSubgroupsofCG（S） 22.5 CartanSubgroups：Summary 23.NormalizerTheorem 23.1 StatementoftheTheorem 23.2 ProofoftheTheorem 23.3 TheVarietyG/B 23.4 Summary IX.CentralizersofTori 24.RegularandSingularTori 24.1 WeylGroups 24.2 RegularTori 24.3 SingularToriandRoots 24.4 Regular1-ParameterSubgroups 25.ActionofaMaximalTorusonG/B 25.1 Actionofa1-ParameterSubgroup 25.2 ExistenceofEnoughFixedPoints 25.3 GroupsofSemisimpleRank1 25.4 WeylChambers 26.TheUnipotentRadical 26.1 CharacterizationofRu（G） 26.2 SomeConsequences 26.3 TheGroupsUa X.StructureofReductiveGroups 27.TheRootSystem 27.1 AbstractRootSystems 27.2 TheIntegralityAxiom 27.3 SimpleRoots 27.4 TheAutomorphismGroupofaSemisimpleGroup 27.5 SimpleComponents 28.BruhatDecomposition 28.1 T-StableSubgroupsofBu 28.2 GroupsofSemisimpleRank1 28.3 TheBruhatDecomposition 28.4 NormalForminG 28.5 Complements 29.TitsSystems 29.1 Axioms 29.2 BruhatDecomposition 29.3 ParabolicSubgroups 29.4 GeneratorsandRelationsforW 29.5 NormalSubgroupsofG 30.ParabolicSubgroups 30.1 StandardParabolicSubgroups 30.2 LeviDecompositions 30.3 ParabolicSubgroupsAssociatedtoCertainUnipotentGroups 30.4 MaximalSubgroupsandMaximalUnipotentSubgroups XI.RepresentationsandClassificationofSemisimpleGroups 31.Representations 31.1 Weights 31.2 MaximalVectors 31.3 IrreducibleRepresentations 31.4 ConstructionofIrreducibleRepresentations 31.5 MultiplicitiesandMinimalHighestWeights 31.6 ContragredientsandInvariantBilinearForms 32.IsomorphismTheorem 32.1 TheClassificationProblem 32.2 ExtensionofψTtoN（T） 32.3 ExtensionofψTtoZa 32.4 ExtensionofψTtoTUa 32.5 ExtensionofψTtoB 32.6 Multiplicativityofψ 33.RootSystemsofRank2 33.1 Reformulationof（A），（B），（C） 33.2 SomePreliminaries 33.3 TypeA2 33.4 TypeB2 33.5 TypeG2 33.6 TheExistenceProblem XII.SurveyofRationalityProperties 34.FieldsofDefinition 34.1 Foundations 34.2 ReviewofEarlierChapters 34.3 Tori 34.4 SomeBasicTheorems 34.5 Borei-TitsStructureTheory 34.6 AnExample：OrthogonalGroups 35.SpecialCases 35.1 SplitandQuasisplitGroups 35.2 FiniteFields 35.3 TheRealField 35.4 LocalFields 35.5 Classification Appendix.RootSystems Bibliography IndexofTerminology IndexofSymbols |
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