SL2(R)
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 examp...
Main Author: | |
---|---|
Format: | eBook |
Language: | English |
Published: |
New York, NY
Springer New York
1985, 1985
|
Edition: | 1st ed. 1985 |
Series: | Graduate Texts in Mathematics
|
Subjects: | |
Online Access: | |
Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
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 3/2
- 8 The equation
- 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