Author | Lang, Serge. author |
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Title | SL 2(R) [electronic resource] / by Serge Lang |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1985 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5142-2 |
Descript | XIV, 431 p. online resource |
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 $$ - \psi ''(y) = {\text{ }}\frac{{s(1 - s)}}{{{ŷ2}}}\psi (y)\;on\;\left[ {a,\infty } \right) $$ -- 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 Schurโs 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