代数几何．4：线性代数群，不变量理论

Introduction

HistoricalComments

Chapter1.LinearAlgebraicGroupsoveranAlgebraically

ClosedField

1.RecollectionsfromAlgebraicGeometry

1.1.AffineVarieties

1.2.Morphisms

1.3.SomeTopologicalProperties

1.4.TangentSpaces

1.5.PropertiesofMorphisms

1.6.Non-AffineVarieties

2.LinearAlgebraicGroups,BasicDefinitionsandProperties

2.1.TheDefinitionofaLinearAlgebraicGroup

2.2.SomeBasicFacts

2.3.G-Spaces

2.4.TheLieAlgebraofanAlgebraicGroup

2.5.Quotients

3.StructuralPropertiesofLinearAlgebraicGroups

3.1.JordanDecompositionandRelatedResults

3.2.DiagonalizableGroupsandTori

3.3.One-DimensionalConnectedGroups

3.4.ConnectedSolvableGroups

3.5.ParabolicSubgroupsandBorelSubgroups

3.6.Radicals,Semi-simpleandReductiveGroups

4.ReductiveGroups

4.1.GroupsofRankOne

4.2.TheRootDatumandtheRootSystem

4.3.BasicPropertiesofReductiveGroups

4.4.ExistenceandUniquenessTheoremsforReductiveGroups

4.5.ClassificationofQuasi-simpleLinearAlgebraicGroups

4.6.RepresentationTheory

Chapter2.LinearAlgebraicGroupsoverArbitraryGroundFields

1.RecollectionsfromAlgebraicGeometry

1.1.F-StructuresonAffineVarieties

1.2.F-StructuresonArbitraryVarieties

1.3.Forms

1.4.RestrictionoftheGroundField

2.F-Groups,BasicProperties

2.1.GeneralitiesAboutF-Groups

2.2.Quotients

2.3.Forms

2.4.RestrictionoftheGroundField

3.Tori

3.1.F-Tori

3.2.F-ToriinF-Groups

3.3.SplitToriinF-Groups

4.SolvableGroups

4.1.SolvableGroups

4.2.Sections

4.3.ElementaryUnipotentGroups

4.4.PropertiesofSplitSolvableGroups

4.5.BasicResultsAboutSolvableF-Groups

5.ReductiveGroups

5.1.SplitReductiveGroups

5.2.ParabolicSubgroups

5.3.TheSmallRootSystem

5.4.TheGroupsG(F)

5.5.TheSphericalTitsBuildingofaReductiveF-Group

6.ClassificationofReductiveF-Groups

6.1.IsomorphismTheorem

6.2.Existence

6.3.RepresentationTheoryofF-Groups

Chapter3.SpecialFields

1.LieAlgebrasofAlgebraicGroupsinCharacteristicZero

1.1.AlgebraicSubalgebras

2.AlgebraicGroupsandLieGroups

2.1.LocallyCompactFields

2.2.RealLieGroups

3.LinearAlgebraicGroupsoverFiniteFields

3.1.LangsTheoremanditsConsequences

3.2.FiniteGroupsofLieType

3.3.RepresentationsofFiniteGroupsofLieType

4.LinearAlgebraicGroupsoverFieldswithaValuation

4.1.TheApartmentandAffineDynkinDiagram

4.2.TheAffineBuilding

4.3.TitsSystem,Decompositions

4.4.LocalFields

5.GlobalFields

5.1.AdeleGroups

5.2.ReductionTheory

5.3.FinitenessResults

5.4.GaloisCohomology

References

　　Thisbookcontainstwocontributionsoncloselyrelatedsubjects:thetheoryoflinearalgebraicgroupsandinvarianttheory.ThefirstpartiswrittenbyT.A.Springer,awell-knownexpertinthefirstmentionedfield.Hcpresentsacomprehensivesurvey,whichcontainsnumeroussketchedproofsandhediscussestheparticularfeaturesofalgebraicgroupsoverspecialfields(finite,local,andglobal).Theauthorsofparttwo-E.B.VinbcrgandV.L.Popov-arcamongthemostactiveresearchersininvarianttheory.Thelast20yearshavebccnaperiodofvigorousdevelopmentinthisfieldductotheinfluenceofmodernmethodsfromalgebraicgeometry.Thebookwillbcveryusefulasareferenceandresearchguidetograduatestudentsandresearchersinmathematicsandtheoreticalphysics.