Table of Contents

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 1.- 4 Symmetry of the Laplace operator on ??.- 5 The Laplace operator on ??.- 6 Green’s functions and the Whittaker equation.- 7 Decomposition of the resolvant on ?? for ? > 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.