Harmonic analysis on semi-simple Lie groups.
Series Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete ; Bd. 188-189Editor: Berlin, New York, Springer-Verlag, 1972Descripción: 2 v. 24 cmISBN: 0387054685 (New York) (v. 1); 9783642516429 (v.2)Tema(s): Semisimple Lie groups | Representations of Lie groups | Harmonic analysisOtra clasificación: 22E45 (22E30)Vol.1 Chapter [1] -- The Structure of Real Semi-Simple Lie Groups -- Introduction [1] -- 1.1 Preliminaries [3] -- 1.1.1 The Structure of Complex Semi-Simple Lie Algebras [3] -- 1.1.2 Root Systems I - Basic Properties [8] -- 1.1.3 Root Systems II - cr-Systems [21] -- 1.1.4 Structure of the Nilpotent Constituent in an Iwasawa Decomposition [34] -- 1.1.5 Reductive Lie Algebras and Groups [41] -- 1.2 The Bruhat Decomposition — Parabolic Subgroups [45] -- 1.2.1 Tits Systems [45] -- 1.2.2 The Complex Case [51] -- 1.2.3 Boundary Subgroups and Parabolic Subgroups of a Real SemiSimple Lie Group [55] -- 1.2.4 Levi Subgroups of a Parabolic Subgroup. The Langlands Decomposition [70] -- 1.3 Cartan Subalgebras [88] -- 1.3.1 Conjugacy of Cartan Subalgebras in a Real Reductive Lie Algebra [88] -- 1.3.2 Classification of Roots [96] -- 1.3.3 Fundamental Cartan Subalgebras [98] -- 1.3.4 Regular and Semi-Regular Elements in a Reductive Lie Algebra [100] -- 1.3.5 Semi-Simple and Nilpotent Elements in a Reductive Lie Algebra [104] -- 1.4 Cartan Subgroups [108] -- 1.4.1 Structure Theorems [108] -- 1.4.2 The Groups W(G,J) and FF(G,J0) [114] -- 1.4.3 Semi-Simple and Unipotent Elements in a Reductive Lie Group [119] -- Chapter [2] -- The Universal Enveloping Algebra of a Semi-Simple Lie Algebra -- Introduction [123] -- 2.1 Invariant Theory I — Generalities [124] -- 2.1.1 Modules [124] -- 2.1.2 The Fundamental Theorem of Invariant Theory [129] -- 2. 1 .3 Invariants or Finite Groups Generated by Reflections [134] -- 2.1.4 Symmetric Algebras and Formal Power Series [140] -- 2.1.5 WeyI Group Invariants [142] -- 2.2 invariant Theory II — Applications to Reductive Lie Algebras [149] -- 2.2.1 A Theorem of Harish-Chandra [149] -- 2.2.2 Theorems of Finitude [152] -- 23 On the Structure of the Universal Enveloping Algebra [159] -- 2.3.1 Generalities [159] -- 2.3.2 Existence of Sufficiently Many Finite Dimensional Representations [163] -- 2.3.3 The Reductive Case [165] -- 2.4 Representations of a Reductive Lie Algebra [170] -- 2.4.1 Simple Modules - The Theorem of Highest Weight [170] -- 2.4.2 The Formula of H. Weyl and B. Kostant [173] -- 2.4.3 The Characters of a Reductive Lie Algebra [176] -- 2.5 Representations on Cohomology Groups [179] -- 2.5.1 The Riemann-Roch Theorem for Lie Algebras [179] -- 2.5.2 Theorems of Bott and Kostant [.183] -- Chapter [3] -- Finite Dimensional Representations of a Semi-Simple Lie Group -- Introduction [192] -- 3.1 Holomorphic Representations of a Complex Semi-Simple Lie Group [193] -- 3.1.1 Generalities [193] -- 3.1.2 The Borel-Weil Theorem [198] -- 3.2 Unitary Representations of a Compact Semi-Simple Lie Group [203] -- 3.2.1 The Invariant Integral on a Compact Semi-Simple Lie Algebra [203] -- 3.2.2 The Plancherel Theorem for a Compact Connected Semi-Simple Lie Group [205] -- 33 Finite Dimensional Class One Representations of a Real Semi-Simple Lie Group [209] -- 3.3.1 The Theorem of E. Cartan and S. Helgason [209] -- 3.3.2 Inequalities [213] -- Chapter [4] -- Infinite Dimensional Group Representation Theory -- 217 Introduction -- 4.1 Representations on a Locally Convex Space [218] -- 4.1.1 Basic Concepts [218] -- 4.1.2 Operations on Representations [223] -- 4.1.3 Intertwining Forms and Operators [226] -- 4.2 Representations on a Banach Space [227] -- 4.2.1 Banach Representations of Associative Algebras [227] -- 4.2.2 Banach Representations of Groups [237] -- 4.3 Representations on a Hilbert Space [243] -- 4.3.1 Generalities [243] -- 4.3.2 Examples [249] -- 4.4 Differentiable Vectors, Analytic Vectors [252] -- 4.4.1 Passage to Ux [252] -- 4.4.2 Absolute Convergence of the Fourier Series [260] -- 4.4.3 A Density Theorem. Fourier Series in Function Spaces [263] -- 4.4.4 Elliptic Elements in the Enveloping Algebra [265] -- 4.4.5 Density of Analytic Vectors - The Theorem of Nelson [276] -- 4.4.6 Analytic Domination - Applications to Representation Theory [290] -- 4.4.7 The Paley-Wiener Space [201] -- 4.5 Large Compact Subgroups [304] -- 4.5.1 The Algebras Ce 4(C), Ze l(G) [304] -- 4.5.2 Groups with Large Compact Subgroups [313] -- 4.5.3 Properties of Largeness [321] -- 4.5.4 Naimark Equivalence [323] -- 4.5.5 Infinitesimal Equivalence [326] -- 4.5.6 Jordan-Holder Series - Multiplicities [332] -- 4.5.7 Theorems of Finitude [335] -- 4.5.8 Characters [342] -- 4.5.9 Square Integrable Representations [349] -- Chapter [5] -- Induced Representations -- Introduction [360] -- 5.1 Unitarily Induced Representations [365] -- 5.1.1 The Definition [365] -- 5.1.2 Unitarily Induced Representations and Measures of Positive Type [383] -- 5. L3 Elementary Properties of Unitarily Induced Representations [385] -- 5.2 Quasi-Invariant Distributions [387] -- 5.2.1 The Global Situation [387] -- 5.2.2 The Local Situation [393] -- 5.2.3 A Fundamental Estimate [395] -- 5.2.4 The Case of Countably Many Orbits [398] -- 5.3 Irreducibility of Unitarily Induced Representations [402] -- 5.3.1 On the Notion of Induced Representation [402] -- 5.3.2 Estimation of the Intertwining Number [403] -- 5.3.3 Reciprocity Theorems [430] -- 5.3.4 Decomposition Theorems [432] -- 5.4 Systems of Imprimitivity [436] -- 5.4.1 Mackey’s Orbital Analysis [436] -- 5.4.2 Examples [440] -- 5.5 Applications to Semi-Simple Lie Groups [444] -- 5.5.1 The Subquotient Theorem [444] -- 5.5.2 Irreducibility of the Principal P-Series - P Minimal [459] -- 5.5.3 The Characters of the Principal P-Series - P Minimal [453] -- 5.5.4 The Riemann-Lebesgue Lemma for the Principal P-Series -P Minimal [471] -- Appendices -- 1 Quasi-Invariant Measures [474] -- 2 Distributions on a Manifold [479] -- 2.1 Differential Operators and Function Spaces [479] -- 2.2 Tensor Products of Topological Vector Spaces [482] -- 2.3 Vector Distributions [486] -- 2.4 Distributions on a Lie Group [488] -- General Notational Conventions [492] -- List of Notations [494] -- Guide to the Literature [497] -- Bibliography [500] -- Subject Index [524] --
Vol. 2 Chapter [6] -- Spherical Functions — The General Theory -- Introduction [1] -- 6.1 Fundamentals [2] -- 6.1.1 Spherical Functions - Functional Properties [2] -- 6.1.2 Spherical Functions - Differential Properties [16] -- 6.2 Examples [20] -- 6.2.1 Spherical Functions on Motion Groups [20] -- 6.2.2 Spherical Functions on Semi-Simple Lie Groups [30] -- Chapter [7] -- Topology on the Dual Plancherel Measure -- Introduction [44] -- 7.1 Topology on the Dual [44] -- 7.1.1 Generalities [44] -- 7.1.2 Applications to Semi-Simple Lie Groups [49] -- 7.2 Plancherel Measure [52] -- 7.2.1 Generalities 1 [52] -- 7.2.2 The Plancherel Theorem for Complex Connected Semi-Simple Lie Groups [54] -- Chapter [8] -- Analysis on a Semi-Simple Lie Group -- Introduction [58] -- 8.1* Preliminaries [59] -- 8.1.1 Acceptable Groups [59] -- 8.1.2 Normalization of Invariant Measures [63] -- 8.1.3 Integration Formulas [67] -- 8.1.4 A Theorem of Compacity [74] -- 8.1.5 The Standard Semi-Norm on a Semi-Simple Lie Group [78] -- 8.1.6 Completely Invariant Sets [80] -- 8.2 Differential Operators on Reductive Lie Groups and Algebras [83] -- 8.2.1 Radial Components of Differential Operators on a Manifold [83] -- 8.2.2 Radial Components of Polynomial Differential Operators on a -- Reductive Lie Algebra [93] -- 8.2.3 Radial Components of Left Invariant Differential Operators on a -- Reductive Lie Group [103] -- 8.2.4 The Connection between Differential Operators in the Algebra and -- on the Group [112] -- 8.3 Central Eigendistributions on Reductive Lie Algebras and Groups [115] -- 8.3.1 The Main Theorem in the Algebra [115] -- 8.3.2 Properties of FT -1 [122] -- 8.3.3 The Main Theorem on the Group [132] -- 8.3.4 Properties of FT - II -- 8.3.5 Rapidly Decreasing Functions on a Euclidean Space [144] -- 8.3.6 Tempered Distributions on a Reductive Lie Algebra [149] -- 8.3.7 Rapidly Decreasing Functions on a Reductive Lie Group [152] -- 8.3.8 Tempered Distributions on a Reductive Lie Group [166] -- 8.3.9 Tools for Harmonic Analysis on G [175] -- 8.4 The Invariant Integral on a Reductive Lie Algebra [178] -- 8.4.1 The Invariant Integral - Definition and Properties [178] -- 8.4.2 Computations in si(2, R) [182] -- 8.4.3 Continuity of the Map [190] -- 8.4.4 Extension Problems [202] -- 8.4.5 The Main Theorem [211] -- 8.5 The Invariant Integral on a Reductive Lie Group [226] -- 8.5.1 The Invariant Integral - Definition and Properties [226] -- 8.5.2 The Inequalities of Descent [236] -- 8.5.3 The Transformations of Descent [242] -- 8.5.4 The Invariant Integral and the Transformations of Descent [245] -- 8.5.5 Estimation of 8.5.6 An Important Inequality [255] -- 8.5.7 Convergence of Certain Integrals [258] -- 8.5.8 Continuity of the Map [261] -- Chapter [9] -- Spherical Functions on a Semi-Simple Lie Group -- Introduction [264] -- 9.1 Asymptotic Behavior of/r-Spherical Functions on a Semi-Simple Lie Group [265] -- 9.1.1 The Main Results [265] -- 9.1.2 Analysis in the Universal Enveloping Algebra [265] -- 9.1.3 The Space ®(/4,x) [281] -- 9.1.4 The Rational Functions TA [286] -- 9.1.5 The Expansion of ^-Spherical Functions [300] -- 9.1.6 Investigation of the c-Function [317] -- 9.1.7 Applications to Zonal Spherical Functions [325] -- 9 J Zonal Spherical Functions on a Semi-Simple Lie Group [335] -- 9.2.1 Statement of Results - Immediate Applications [335] -- 9.2.2 The Plancherel Theorem for P(G) [335] -- 9.2.3 The Paley-Wiener Theorem for/’(G) [344] -- 9.2.4 Harmonic Analysis in P(G) [353] -- 9.3 Spherical Functions and Differential Equations [367] -- 9.3.1 The Weak Inequality and Some of its Implications [367] -- 9.3.2 Existence and Uniqueness of the Indices I [372] -- 9.3.3 Existence and Uniqueness of the Indices II [377] -- Chapter [10] -- The Discrete Series for a Semi-Simple Lie Group — -- Existence and Exhaustion -- Introduction [389] -- 10.1 The Role of the Distributions €>, in the Harmonic Analysis on G [390] -- 10.1.1 Existence and Uniqueness of the 0, [390] -- 10.1.2 Expansion of 3-Finite Functions in ^(G) [397] -- 10.2 Theory of the Discrete Series [400] -- 10.2.1 Existence of the Discrete Series [400] -- 10.2.2 The Characters of the Discrete Series I - Implication of the -- Orthogonality Relations [401] -- 10.2.3 The Characters of the Discrete Series II - Application of the -- Differential Equations [404] -- 10.2.4 The Theorem of Harish-Chandra. [407] -- Epilogue [414] -- Appendix -- 3 Some Results on Differential Equations [426] -- 3.1 The Main Theorems [426] -- 3.2 Lemmas from Analysis [428] -- 3.3 Analytic Continuation of Solutions [430] -- 3.4 Decent Convergence [432] -- 3.5 Normal Sequences of f-Polynomials [433] -- General Notational Conventions [450] -- List of Notations [452] -- Juide to the Literature [456] -- Bibliography [460] -- Subject Index to Volumes I and II [484] -- Chapter 1 The Structure of Real Semi-Simple Lie Groups -- Chapter 2 The Universal Enveloping Algebra of a Semi-Simple Lie Algebra -- Chapter 3 Finite Dimensional Representations of a Semi-Simple Lie Group -- Chapter 4 Infinite Dimensional Group Representation Theory -- Chapter 5 Induced Representations -- Appendix 1 Quasi-Invariant Measures -- Appendix 2 Distributions on a Manifold --
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Includes bibliographies.
Vol.1
Chapter [1] --
The Structure of Real Semi-Simple Lie Groups --
Introduction [1] --
1.1 Preliminaries [3] --
1.1.1 The Structure of Complex Semi-Simple Lie Algebras [3] --
1.1.2 Root Systems I - Basic Properties [8] --
1.1.3 Root Systems II - cr-Systems [21] --
1.1.4 Structure of the Nilpotent Constituent in an Iwasawa Decomposition [34] --
1.1.5 Reductive Lie Algebras and Groups [41] --
1.2 The Bruhat Decomposition — Parabolic Subgroups [45] --
1.2.1 Tits Systems [45] --
1.2.2 The Complex Case [51] --
1.2.3 Boundary Subgroups and Parabolic Subgroups of a Real SemiSimple Lie Group [55] --
1.2.4 Levi Subgroups of a Parabolic Subgroup. The Langlands Decomposition [70] --
1.3 Cartan Subalgebras [88] --
1.3.1 Conjugacy of Cartan Subalgebras in a Real Reductive Lie Algebra [88] --
1.3.2 Classification of Roots [96] --
1.3.3 Fundamental Cartan Subalgebras [98] --
1.3.4 Regular and Semi-Regular Elements in a Reductive Lie Algebra [100] --
1.3.5 Semi-Simple and Nilpotent Elements in a Reductive Lie Algebra [104] --
1.4 Cartan Subgroups [108] --
1.4.1 Structure Theorems [108] --
1.4.2 The Groups W(G,J) and FF(G,J0) [114] --
1.4.3 Semi-Simple and Unipotent Elements in a Reductive Lie Group [119] --
Chapter [2] --
The Universal Enveloping Algebra of a Semi-Simple Lie Algebra --
Introduction [123] --
2.1 Invariant Theory I — Generalities [124] --
2.1.1 Modules [124] --
2.1.2 The Fundamental Theorem of Invariant Theory [129] --
2. 1 .3 Invariants or Finite Groups Generated by Reflections [134] --
2.1.4 Symmetric Algebras and Formal Power Series [140] --
2.1.5 WeyI Group Invariants [142] --
2.2 invariant Theory II — Applications to Reductive Lie Algebras [149] --
2.2.1 A Theorem of Harish-Chandra [149] --
2.2.2 Theorems of Finitude [152] --
23 On the Structure of the Universal Enveloping Algebra [159] --
2.3.1 Generalities [159] --
2.3.2 Existence of Sufficiently Many Finite Dimensional Representations [163] --
2.3.3 The Reductive Case [165] --
2.4 Representations of a Reductive Lie Algebra [170] --
2.4.1 Simple Modules - The Theorem of Highest Weight [170] --
2.4.2 The Formula of H. Weyl and B. Kostant [173] --
2.4.3 The Characters of a Reductive Lie Algebra [176] --
2.5 Representations on Cohomology Groups [179] --
2.5.1 The Riemann-Roch Theorem for Lie Algebras [179] --
2.5.2 Theorems of Bott and Kostant [.183] --
Chapter [3] --
Finite Dimensional Representations of a Semi-Simple Lie Group --
Introduction [192] --
3.1 Holomorphic Representations of a Complex Semi-Simple Lie Group [193] --
3.1.1 Generalities [193] --
3.1.2 The Borel-Weil Theorem [198] --
3.2 Unitary Representations of a Compact Semi-Simple Lie Group [203] --
3.2.1 The Invariant Integral on a Compact Semi-Simple Lie Algebra [203] --
3.2.2 The Plancherel Theorem for a Compact Connected Semi-Simple Lie Group [205] --
33 Finite Dimensional Class One Representations of a Real Semi-Simple Lie Group [209] --
3.3.1 The Theorem of E. Cartan and S. Helgason [209] --
3.3.2 Inequalities [213] --
Chapter [4] --
Infinite Dimensional Group Representation Theory --
217 Introduction --
4.1 Representations on a Locally Convex Space [218] --
4.1.1 Basic Concepts [218] --
4.1.2 Operations on Representations [223] --
4.1.3 Intertwining Forms and Operators [226] --
4.2 Representations on a Banach Space [227] --
4.2.1 Banach Representations of Associative Algebras [227] --
4.2.2 Banach Representations of Groups [237] --
4.3 Representations on a Hilbert Space [243] --
4.3.1 Generalities [243] --
4.3.2 Examples [249] --
4.4 Differentiable Vectors, Analytic Vectors [252] --
4.4.1 Passage to Ux [252] --
4.4.2 Absolute Convergence of the Fourier Series [260] --
4.4.3 A Density Theorem. Fourier Series in Function Spaces [263] --
4.4.4 Elliptic Elements in the Enveloping Algebra [265] --
4.4.5 Density of Analytic Vectors - The Theorem of Nelson [276] --
4.4.6 Analytic Domination - Applications to Representation Theory [290] --
4.4.7 The Paley-Wiener Space [201] --
4.5 Large Compact Subgroups [304] --
4.5.1 The Algebras Ce 4(C), Ze l(G) [304] --
4.5.2 Groups with Large Compact Subgroups [313] --
4.5.3 Properties of Largeness [321] --
4.5.4 Naimark Equivalence [323] --
4.5.5 Infinitesimal Equivalence [326] --
4.5.6 Jordan-Holder Series - Multiplicities [332] --
4.5.7 Theorems of Finitude [335] --
4.5.8 Characters [342] --
4.5.9 Square Integrable Representations [349] --
Chapter [5] --
Induced Representations --
Introduction [360] --
5.1 Unitarily Induced Representations [365] --
5.1.1 The Definition [365] --
5.1.2 Unitarily Induced Representations and Measures of Positive Type [383] --
5. L3 Elementary Properties of Unitarily Induced Representations [385] --
5.2 Quasi-Invariant Distributions [387] --
5.2.1 The Global Situation [387] --
5.2.2 The Local Situation [393] --
5.2.3 A Fundamental Estimate [395] --
5.2.4 The Case of Countably Many Orbits [398] --
5.3 Irreducibility of Unitarily Induced Representations [402] --
5.3.1 On the Notion of Induced Representation [402] --
5.3.2 Estimation of the Intertwining Number [403] --
5.3.3 Reciprocity Theorems [430] --
5.3.4 Decomposition Theorems [432] --
5.4 Systems of Imprimitivity [436] --
5.4.1 Mackey’s Orbital Analysis [436] --
5.4.2 Examples [440] --
5.5 Applications to Semi-Simple Lie Groups [444] --
5.5.1 The Subquotient Theorem [444] --
5.5.2 Irreducibility of the Principal P-Series - P Minimal [459] --
5.5.3 The Characters of the Principal P-Series - P Minimal [453] --
5.5.4 The Riemann-Lebesgue Lemma for the Principal P-Series -P Minimal [471] --
Appendices --
1 Quasi-Invariant Measures [474] --
2 Distributions on a Manifold [479] --
2.1 Differential Operators and Function Spaces [479] --
2.2 Tensor Products of Topological Vector Spaces [482] --
2.3 Vector Distributions [486] --
2.4 Distributions on a Lie Group [488] --
General Notational Conventions [492] --
List of Notations [494] --
Guide to the Literature [497] --
Bibliography [500] --
Subject Index [524] --
Vol. 2
Chapter [6] --
Spherical Functions — The General Theory --
Introduction [1] --
6.1 Fundamentals [2] --
6.1.1 Spherical Functions - Functional Properties [2] --
6.1.2 Spherical Functions - Differential Properties [16] --
6.2 Examples [20] --
6.2.1 Spherical Functions on Motion Groups [20] --
6.2.2 Spherical Functions on Semi-Simple Lie Groups [30] --
Chapter [7] --
Topology on the Dual Plancherel Measure --
Introduction [44] --
7.1 Topology on the Dual [44] --
7.1.1 Generalities [44] --
7.1.2 Applications to Semi-Simple Lie Groups [49] --
7.2 Plancherel Measure [52] --
7.2.1 Generalities 1 [52] --
7.2.2 The Plancherel Theorem for Complex Connected Semi-Simple Lie Groups [54] --
Chapter [8] --
Analysis on a Semi-Simple Lie Group --
Introduction [58] --
8.1* Preliminaries [59] --
8.1.1 Acceptable Groups [59] --
8.1.2 Normalization of Invariant Measures [63] --
8.1.3 Integration Formulas [67] --
8.1.4 A Theorem of Compacity [74] --
8.1.5 The Standard Semi-Norm on a Semi-Simple Lie Group [78] --
8.1.6 Completely Invariant Sets [80] --
8.2 Differential Operators on Reductive Lie Groups and Algebras [83] --
8.2.1 Radial Components of Differential Operators on a Manifold [83] --
8.2.2 Radial Components of Polynomial Differential Operators on a --
Reductive Lie Algebra [93] --
8.2.3 Radial Components of Left Invariant Differential Operators on a --
Reductive Lie Group [103] --
8.2.4 The Connection between Differential Operators in the Algebra and --
on the Group [112] --
8.3 Central Eigendistributions on Reductive Lie Algebras and Groups [115] --
8.3.1 The Main Theorem in the Algebra [115] --
8.3.2 Properties of FT -1 [122] --
8.3.3 The Main Theorem on the Group [132] --
8.3.4 Properties of FT - II --
8.3.5 Rapidly Decreasing Functions on a Euclidean Space [144] --
8.3.6 Tempered Distributions on a Reductive Lie Algebra [149] --
8.3.7 Rapidly Decreasing Functions on a Reductive Lie Group [152] --
8.3.8 Tempered Distributions on a Reductive Lie Group [166] --
8.3.9 Tools for Harmonic Analysis on G [175] --
8.4 The Invariant Integral on a Reductive Lie Algebra [178] --
8.4.1 The Invariant Integral - Definition and Properties [178] --
8.4.2 Computations in si(2, R) [182] --
8.4.3 Continuity of the Map [190] --
8.4.4 Extension Problems [202] --
8.4.5 The Main Theorem [211] --
8.5 The Invariant Integral on a Reductive Lie Group [226] --
8.5.1 The Invariant Integral - Definition and Properties [226] --
8.5.2 The Inequalities of Descent [236] --
8.5.3 The Transformations of Descent [242] --
8.5.4 The Invariant Integral and the Transformations of Descent [245] --
8.5.5 Estimation of 8.5.6 An Important Inequality [255] --
8.5.7 Convergence of Certain Integrals [258] --
8.5.8 Continuity of the Map [261] --
Chapter [9] --
Spherical Functions on a Semi-Simple Lie Group --
Introduction [264] --
9.1 Asymptotic Behavior of/r-Spherical Functions on a Semi-Simple Lie Group [265] --
9.1.1 The Main Results [265] --
9.1.2 Analysis in the Universal Enveloping Algebra [265] --
9.1.3 The Space ®(/4,x) [281] --
9.1.4 The Rational Functions TA [286] --
9.1.5 The Expansion of ^-Spherical Functions [300] --
9.1.6 Investigation of the c-Function [317] --
9.1.7 Applications to Zonal Spherical Functions [325] --
9 J Zonal Spherical Functions on a Semi-Simple Lie Group [335] --
9.2.1 Statement of Results - Immediate Applications [335] --
9.2.2 The Plancherel Theorem for P(G) [335] --
9.2.3 The Paley-Wiener Theorem for/’(G) [344] --
9.2.4 Harmonic Analysis in P(G) [353] --
9.3 Spherical Functions and Differential Equations [367] --
9.3.1 The Weak Inequality and Some of its Implications [367] --
9.3.2 Existence and Uniqueness of the Indices I [372] --
9.3.3 Existence and Uniqueness of the Indices II [377] --
Chapter [10] --
The Discrete Series for a Semi-Simple Lie Group — --
Existence and Exhaustion --
Introduction [389] --
10.1 The Role of the Distributions €>, in the Harmonic Analysis on G [390] --
10.1.1 Existence and Uniqueness of the 0, [390] --
10.1.2 Expansion of 3-Finite Functions in ^(G) [397] --
10.2 Theory of the Discrete Series [400] --
10.2.1 Existence of the Discrete Series [400] --
10.2.2 The Characters of the Discrete Series I - Implication of the --
Orthogonality Relations [401] --
10.2.3 The Characters of the Discrete Series II - Application of the --
Differential Equations [404] --
10.2.4 The Theorem of Harish-Chandra. [407] --
Epilogue [414] --
Appendix --
3 Some Results on Differential Equations [426] --
3.1 The Main Theorems [426] --
3.2 Lemmas from Analysis [428] --
3.3 Analytic Continuation of Solutions [430] --
3.4 Decent Convergence [432] --
3.5 Normal Sequences of f-Polynomials [433] --
General Notational Conventions [450] --
List of Notations [452] --
Juide to the Literature [456] --
Bibliography [460] --
Subject Index to Volumes I and II [484] --
Chapter 1 The Structure of Real Semi-Simple Lie Groups --
Chapter 2 The Universal Enveloping Algebra of a Semi-Simple Lie Algebra --
Chapter 3 Finite Dimensional Representations of a Semi-Simple Lie Group --
Chapter 4 Infinite Dimensional Group Representation Theory --
Chapter 5 Induced Representations --
Appendix 1 Quasi-Invariant Measures --
Appendix 2 Distributions on a Manifold --
MR,
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