I General Results. - 1 The representation on Cc(G). - 2 A criterion for complete reducibility. - 3 L2 kernels and operators. - 4 Plancherel measures. - II Compact Groups. - 1 Decomposition over K for SL2(R). - 2 Compact groups in general. - III Induced Representations. - 1 Integration on coset spaces. - 2 Induced representations. - 3 Associated spherical functions. - 4 The kernel defining the induced representation. - IV Spherical Functions. - 1 Bi-invariance. - 2 Irreducibility. - 3 The spherical property. - 4 Connection with unitary representations. - 5 Positive definite functions. - V The Spherical Transform. - 1 Integral formulas. - 2 The Harish transform. - 3 The Mellin transfor. - 4 The spherical transform. - 5 Explicit formulas and asymptotic expansions. - 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. - 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. - VIII The PlanchereS Formula. - 1 Calculus lemma. - 2 The Harish transforms discontinuities. - 3 Some lemmas. - 4 The Plancherel formula. - 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. - X Partial Differential Operators. - 1 The universal enveloping algebra. - 2 Analytic vectors. - 3 Eigenfunctions of ?f. - XI The Well Representation. - 1 3/2. - 8 The equation $$ - \psi ''(y) = {\text{ }}\frac{{s(1 - s)}}{{{y^2}}}\psi (y)\;on\;\left[ {a\infty } \right) $$. - 9 Eigenfunctions of the Laplacian in L2?\? = H. - 10 The resolvant equations for 0< ? < 2. - 11 The kernel of the resolvant for 0 < ? < 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 PDF. - 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. - Symbols Frequently Used. Language: English