Real Productive Groups I, Vol 132. Pure and Applied Mathematics

  • ID: 2986242
  • Book
  • 412 Pages
  • Elsevier Science and Technology
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Real Reductive Groups I is an introduction to the representation theory of real reductive groups. It is based on courses that the author has given at Rutgers for the past 15 years. It also had its genesis in an attempt of the author to complete a manuscript of the lectures that he gave at the CBMS regional conference at The University of North Carolina at Chapel Hill in June of 1981.
This book comprises 10 chapters and begins with some background material as an introduction. The following chapters then discuss elementary representation theory; real reductive groups; the basic theory of (g, K)-modules; the asymptotic behavior of matrix coefficients; The Langlands Classification; a construction of the fundamental series; cusp forms on G; character theory; and unitary representations and (g, K)-cohomology.
This book will be of interest to mathematicians and statisticians.

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Preface

Introduction


Chapter 0. Background Material


Introduction


0.1 Invariant measures on homogeneous spaces


0.2 The structure of reductive Lie algebras


0.3 The structure of compact Lie groups


0.4 The universal enveloping algebra of a Lie algebra


0.5. Some basic representation theory


0.6 Modules over the universal enveloping algebra


Chapter 1. Elementary Representation Theory


Introduction


1.1. General properties of representations


1.2. Schur's lemma


1.3. Square integrable representations


1.4. Basic representation theory of compact 9 groups


1.5. A class of induced representations


1.6. C" vectors and analytic vectors


1.7. Representations of compact Lie groups


1.8. Further results and comments


Chapter 2. Real Reductive Groups


Introduction


2.1. The definition of a real reductive group


2.2. Parabolic pairs


2.3. Cartan subgroups


2.4. Integration formulas


2.5. The Weyl character formula


2.A. Appendices to Chapter 2


2.A.1. Some linear algebra


2.A.2. Norms on real reductive groups


Chapter 3. The Basic Theory of (g, K)-Modules


Introduction


3.1. The Chevalley restriction theorem


3.2. The Harish-Chandra isomorphism of the center of the universal enveloping algebra


3.3. (g, K)-modules


3.4. A basic theorem of Harish-Chandra


3.5. The subquotient theorem


3.6. The spherical principal series


3.7. A Lemma of Osborne


3.8. The subrepresentation theorem


3.9. Notes and further results


3.A. Appendices to Chapter 3


3.A.1. Some associative algebra


3.A.2. A Lemma of Harish-Chandra


Chapter 4. The Asymptotic Behavior of Matrix Coefficients


Introduction


4.1 The Jacquet module of an admissible (g, K)-module


4.2 Three applications of the Jacquet module


4.3 Asymptotic behavior of matrix coefficients


4.4 Asymptotic expansions of matrix coefficients


4.5. Harish-Chandra's E-function


4.6. Notes and further results


4.A. Appendices to Chapter 4


4.A.1. Asymptotic expansions


4.A.2. Some inequalities


Chapter 5. The Langlands Classification


Introduction


5.1. Tempered (g, K)-modules


5.2. The principal series


5.3. The intertwining integrals


5.4. The Langlands classification


5.5. Some applications of the classification


5.6. SL(2,R)


5.7. SL(2,C)


5.8. Notes and further results


5.A. Appendices to Chapter 5


5.A.1. A Lemma of Langlands


5.A.2. An a priori estimate


5.A.3. Square integrability and the polar decomposition


Chapter 6. A Construction of the Fundamental Series


Introduction


6.1 Relative Lie algebra cohomology


6.2 A construction of (f, K)-modules


6.3 The Zuckerman functors


6.4 Some vanishing theorems


6.5 Blattner type formulas


6.6 Irreducibility


6.7 Unitarizability


6.8 Temperedness and square integrability


6.9 The case of disconnected G


6.10 Notes and further results


6.A Appendices to Chapter 6


6.A.1 Some homological algebra


6.A.2 Partition functions


6.A.3 Tensor products with finite dimensional representations


6.A.4 Infinitesimally unitary modules


Chapter 7. Cusp Forms on G


Introduction


7.1. Some Fréchet spaces of functions on G


7.2. The Harish-Chandra transform


7.3. Orbital integrals on a reductive Lie algebra


7.4 Orbital integral on a reductive Lie group


7.5 The orbital integrals of cusp forms


7.6 Harmonic analysis on the space of cusp forms


7.7 Square integrable representations revisited


7.8 Notes and further results


7.A Appendices to Chapter 7


7.A.1 Some linear algebra


7.A.2 Radial components on the Lie algebra


7.A.3 Radial components on the Lie group


7.A.4 Some harmonic analysis on Tori


7.A.5 Fundamental solutions of certain differential operators


Chapter 8. Character Theory


Introduction


8.1 The character of an admissible representation


8.2 The K-character of a (g, K)-module


8.3 Harish-Chandra's regularity theorem on the Lie algebra


8.4 Harish-Chandra's regularity theorem on the Lie group


8.5 Tempered invariant Z(g)-finite distributions on G


8.6. Harish-Chandra's basic inequality


8.7. The completeness of the ?


8.A. Appendices to Chapter 8


8.A.1 Trace class operators


8.A.2. Some operations on distributions


8.A.3. The radial component revisited


8.A.4. The orbit structure on a real reductive Lie algebra


8.A.5. Some technical results for Harish-Chandra's regularity theorem


Chapter 9. Unitary Representations and (g, K)-Cohomology


Introduction


9.1. Tensor products of finite dimensional representations


9.2. Spinors


9.3. The Dirac operator


9.4. (g, K)-cohomology


9.5. Some results of Kumaresan, Parthasarathy, Vogan, Zuckerman


9.6. ?-cohomology


9.7. A theorem of Vogan-Zuckerman


9.8. Further results


9.A. Appendices to Chapter 9


9.A.1. Weyl groups


9.A.2. Spectral sequences


Bibliography


Index
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Wallach, Nolan R.
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