SL2(R)

This book introduces the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example - SL2(R). The contents are accessible to a wide audience, requiring only a knowledge of real analysis, and some differential equations.

SL2(R) gives the student an introduction to the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example - SL2(R). This field is of interest not only for its own sake, but for its connections with other areas such as number theory, as brought out, for example, in the work of Langlands. The rapid development of representation theory over the past 40 years has made it increasingly difficult for a student to enter the field. This book makes the theory accessible to a wide audience, its only prerequisites being a knowledge of real analysis, and some differential equations.

Contenu
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.