Starting with Bargmann's paPer on the tnhmte dimenslonal representattons of SL2(R)，the theory of representations of semisimple Lie groups has evolved toa rather extensive production.

Notation

Chapter Ⅰ General Results

 1 The representation on Cv(G)

 2 A criterion for complete reducibility

 3 L2 kernels and operators

 4 Plancherel measures

Chapter Ⅱ Compact Groups

 1 Decomposition over K for SL2(R)

 2 Compact groups in general

Chapter Ⅲ Induced Representations

 1 Integration on coset spaces

 2 Induced representations

 3 Associated spherical functions

 4 The kernel defining the induced representation

Chapter Ⅳ Spherical Functions

 1 Bi-invariance

 2 Irreducibility

 3 The spherical property

 4 Connection with unitary representations

 5 Positive definite functions

Chapter Ⅴ The Spherical Transform

 1 Integral formulas

 2 The Harish transform

 3 The Mellin transform

 4 The spherical transform

 5 Explicit formulas and asymptotic expansions

Chapter Ⅵ The Derived Representation on the Lie Algebra

 1 The derived representation

 2 The derived representation decomposed over K

 3 Unitarization of a representation

 4 The Lie derivatives on G

 5 Irreducible components of the induced representations

 6 Classification of all unitary irreducible representations

 7 Separation by the trace

Chapter Ⅶ Traces

 1 Operators of trace class

 2 Integral formulas

 3 The trace in the induced representation

 4 The trace in the discrete series

 5 Relation between the Harish transforms on A and K

 Appendix. General facts about traces

Chapter Ⅷ The Planeherel Formula

 1 Calculus lemma

 2 The Harish transforms discontinuities

 3 Some iemmas

 4 The Plancherel formula

Chapter Ⅸ Discrete Series

 1 Discrete series in L2(G)

 2 Representation in the upper half plane

 3 Representation on the disc

 4 The lifting of weight m

 5 The holomorphic property

Chapter Ⅹ Partial Differential Operators

 1 The universal enveloping algebra

 2 Analytic vectors

 3 Eiaenfunctions of Z (f)

Chapter Ⅺ The Well Representation

 1 Some convolutions

 2 Generators and relations for SL2

 3 The Well representation

Chapter Ⅻ Representation on OL2(Г\\G)

 1 Cusps on the group

 2 Cusp forms

 3 A criterion for compact operators

 4 Complete reducibility of OL2Г\\G)

Chapter ⅩⅢ The Continuous Part of L2(Г\\G)

 1 An orthogonality relation

 2 The Eisenstein series

 3 Analytic continuation and functional equation

 4 Mellin and zeta transforms

 5 Some group theoretic lemmas

 6 An expression for TOTφ

 7 Analytic continuation of the zeta transform of TOTφ

 8 The spectral decomposition

Chapter ⅩⅣ Spectral Decomposition of the Laplace Operator on Г\\■

 1 Geometry and differential operators on ■

 2 A solution of ιφ≠s(ι-s)φ

 3 The resoivant of the Laplace operator on ■ for σ＞I

 4 Symmetry of the Laplace operator on Г\\■

 5 The Laplace operator on Г\\■

 6 Green's functions and the Whittaker equation

 7 Decomposition of the resolvant on Г\\■ for o＞3/2

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

 9 Eigenfunctions of the Laplace, an in L2(Г\\■)＝H

 10 The resolvant equations for 0 ＜α＜2

 11 The kernel of the resolvant for 0 ＜α＜ 2

 12 The Eisenstein operator and Eisenstein functions

 13 The continuous part of the spectrum

 14 Several cusps

Appendix 1 Bounded Hermitian Operators and Schur's Lemma

 1 Continuous functions of operators

 2 Projection functions of operators

Appendix 2 Unbounded Operators

 1 Self-adjoint operators

 2 The spectral measure

 3 The resolvant formula

Appendix 3 Meromorphic Families of Operators

 1 Compact operators

 2 Bounded operators

Appendix 4 Elliptic PDE

 1 Sobolev spaces

 2 Ordinary estimates

 3 Elliptic estimates

 4 Compactness and regularity on the torus

 5 Regularity in Euclidean space

Appendix 5 Weak and Strong Analyticity

 1 Complex theorem

 2 Real theorem

Bibliography

Symbols Frequently Used

Index