SL2（R）

Notation

ChapterⅠGeneralResults

1TherepresentationonCv(G)

2Acriterionforcompletereducibility

3L2kernelsandoperators

4Plancherelmeasures

ChapterⅡCompactGroups

1DecompositionoverKforSL2(R)

2Compactgroupsingeneral

ChapterⅢInducedRepresentations

1Integrationoncosetspaces

2Inducedrepresentations

3Associatedsphericalfunctions

4Thekerneldefiningtheinducedrepresentation

ChapterⅣSphericalFunctions

1Bi-invariance

2Irreducibility

3Thesphericalproperty

4Connectionwithunitaryrepresentations

5Positivedefinitefunctions

ChapterⅤTheSphericalTransform

1Integralformulas

2TheHarishtransform

3TheMellintransform

4Thesphericaltransform

5Explicitformulasandasymptoticexpansions

ChapterⅥTheDerivedRepresentationontheLieAlgebra

1Thederivedrepresentation

2ThederivedrepresentationdecomposedoverK

3Unitarizationofarepresentation

4TheLiederivativesonG

5Irreduciblecomponentsoftheinducedrepresentations

6Classificationofallunitaryirreduciblerepresentations

7Separationbythetrace

ChapterⅦTraces

1Operatorsoftraceclass

2Integralformulas

3Thetraceintheinducedrepresentation

4Thetraceinthediscreteseries

5RelationbetweentheHarishtransformsonAandK

Appendix.Generalfactsabouttraces

ChapterⅧThePlaneherelFormula

1Calculuslemma

2TheHarishtransformsdiscontinuities

3Someiemmas

4ThePlancherelformula

ChapterⅨDiscreteSeries

1DiscreteseriesinL2(G)

2Representationintheupperhalfplane

3Representationonthedisc

4Theliftingofweightm

5Theholomorphicproperty

ChapterⅩPartialDifferentialOperators

1Theuniversalenvelopingalgebra

2Analyticvectors

3EiaenfunctionsofZ(f)

ChapterⅪTheWellRepresentation

1Someconvolutions

2GeneratorsandrelationsforSL2

3TheWellrepresentation

ChapterⅫRepresentationonOL2(Г\G)

1Cuspsonthegroup

2Cuspforms

3Acriterionforcompactoperators

4CompletereducibilityofOL2Г\G)

ChapterⅩⅢTheContinuousPartofL2(Г\G)

1Anorthogonalityrelation

2TheEisensteinseries

3Analyticcontinuationandfunctionalequation

4Mellinandzetatransforms

5Somegrouptheoreticlemmas

6AnexpressionforTOTφ

7AnalyticcontinuationofthezetatransformofTOTφ

8Thespectraldecomposition

ChapterⅩⅣSpectralDecompositionoftheLaplaceOperatoronГ\■

1Geometryanddifferentialoperatorson■

2Asolutionofιφ≠s(ι-s)φ

3TheresoivantoftheLaplaceoperatoron■forσ＞I

4SymmetryoftheLaplaceoperatoronГ\■

5TheLaplaceoperatoronГ\■

6GreensfunctionsandtheWhittakerequation

7DecompositionoftheresolvantonГ\■foro＞3/2

8Theequation-ψ″(y)＝s(ι-s)/y2ψ(y)on[α,∞)

9EigenfunctionsoftheLaplace,aninL2(Г\■)＝H

10Theresolvantequationsfor0＜α＜2

11Thekerneloftheresolvantfor0＜α＜2

12TheEisensteinoperatorandEisensteinfunctions

13Thecontinuouspartofthespectrum

14Severalcusps

Appendix1BoundedHermitianOperatorsandSchursLemma

1Continuousfunctionsofoperators

2Projectionfunctionsofoperators

Appendix2UnboundedOperators

1Self-adjointoperators

2Thespectralmeasure

3Theresolvantformula

Appendix3MeromorphicFamiliesofOperators

1Compactoperators

2Boundedoperators

Appendix4EllipticPDE

1Sobolevspaces

2Ordinaryestimates

3Ellipticestimates

4Compactnessandregularityonthetorus

5RegularityinEuclideanspace

Appendix5WeakandStrongAnalyticity

1Complextheorem

2Realtheorem

Bibliography

SymbolsFrequentlyUsed

Index

　　StartingwithBargmannspaPeronthetnhmtedimenslonalrepresentattonsofSL2(R)，thetheoryofrepresentationsofsemisimpleLiegroupshasevolvedtoaratherextensiveproduction.