Real Reductive Groups I: Volume 132 (Pure and Applied Mathematics") 〈132〉

Wallach, Nolan R.

Academic Press（1988/02発売）

• 製本 Hardcover:ハードカバー版／ページ数 412 p.
• 言語 ENG
• 商品コード 9780127329604
• DDC分類 512.55

Full Description

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.

Contents

Preface IntroductionChapter 0. Background MaterialIntroduction0.1 Invariant measures on homogeneous spaces0.2 The structure of reductive Lie algebras0.3 The structure of compact Lie groups0.4 The universal enveloping algebra of a Lie algebra0.5. Some basic representation theory0.6 Modules over the universal enveloping algebraChapter 1. Elementary Representation TheoryIntroduction1.1. General properties of representations1.2. Schur's lemma1.3. Square integrable representations1.4. Basic representation theory of compact 9 groups1.5. A class of induced representations1.6. C" vectors and analytic vectors1.7. Representations of compact Lie groups1.8. Further results and commentsChapter 2. Real Reductive GroupsIntroduction2.1. The definition of a real reductive group2.2. Parabolic pairs2.3. Cartan subgroups2.4. Integration formulas2.5. The Weyl character formula2.A. Appendices to Chapter 22.A.1. Some linear algebra2.A.2. Norms on real reductive groupsChapter 3. The Basic Theory of (g, K)-ModulesIntroduction3.1. The Chevalley restriction theorem3.2. The Harish-Chandra isomorphism of the center of the universal enveloping algebra3.3. (g, K)-modules3.4. A basic theorem of Harish-Chandra3.5. The subquotient theorem3.6. The spherical principal series3.7. A Lemma of Osborne3.8. The subrepresentation theorem3.9. Notes and further results3.A. Appendices to Chapter 33.A.1. Some associative algebra3.A.2. A Lemma of Harish-ChandraChapter 4. The Asymptotic Behavior of Matrix CoefficientsIntroduction4.1 The Jacquet module of an admissible (g, K)-module4.2 Three applications of the Jacquet module4.3 Asymptotic behavior of matrix coefficients4.4 Asymptotic expansions of matrix coefficients4.5. Harish-Chandra's E-function4.6. Notes and further results4.A. Appendices to Chapter 44.A.1. Asymptotic expansions4.A.2. Some inequalitiesChapter 5. The Langlands ClassificationIntroduction5.1. Tempered (g, K)-modules5.2. The principal series5.3. The intertwining integrals5.4. The Langlands classification5.5. Some applications of the classification5.6. SL(2,R)5.7. SL(2,C)5.8. Notes and further results5.A. Appendices to Chapter 55.A.1. A Lemma of Langlands5.A.2. An a priori estimate5.A.3. Square integrability and the polar decompositionChapter 6. A Construction of the Fundamental SeriesIntroduction6.1 Relative Lie algebra cohomology6.2 A construction of (f, K)-modules6.3 The Zuckerman functors6.4 Some vanishing theorems6.5 Blattner type formulas6.6 Irreducibility6.7 Unitarizability6.8 Temperedness and square integrability6.9 The case of disconnected G6.10 Notes and further results6.A Appendices to Chapter 66.A.1 Some homological algebra6.A.2 Partition functions6.A.3 Tensor products with finite dimensional representations6.A.4 Infinitesimally unitary modulesChapter 7. Cusp Forms on GIntroduction7.1. Some Frechet spaces of functions on G7.2. The Harish-Chandra transform7.3. Orbital integrals on a reductive Lie algebra7.4 Orbital integral on a reductive Lie group7.5 The orbital integrals of cusp forms7.6 Harmonic analysis on the space of cusp forms7.7 Square integrable representations revisited7.8 Notes and further results7.A Appendices to Chapter 77.A.1 Some linear algebra7.A.2 Radial components on the Lie algebra7.A.3 Radial components on the Lie group7.A.4 Some harmonic analysis on Tori7.A.5 Fundamental solutions of certain differential operatorsChapter 8. Character TheoryIntroduction8.1 The character of an admissible representation8.2 The K-character of a (g, K)-module8.3 Harish-Chandra's regularity theorem on the Lie algebra8.4 Harish-Chandra's regularity theorem on the Lie group8.5 Tempered invariant Z(g)-finite distributions on G8.6. Harish-Chandra's basic inequality8.7. The completeness of the p8.A. Appendices to Chapter 88.A.1 Trace class operators8.A.2. Some operations on distributions8.A.3. The radial component revisited8.A.4. The orbit structure on a real reductive Lie algebra8.A.5. Some technical results for Harish-Chandra's regularity theoremChapter 9. Unitary Representations and (g, K)-CohomologyIntroduction9.1. Tensor products of finite dimensional representations9.2. Spinors9.3. The Dirac operator9.4. (g, K)-cohomology9.5. Some results of Kumaresan, Parthasarathy, Vogan, Zuckerman9.6. -cohomology9.7. A theorem of Vogan-Zuckerman9.8. Further results9.A. Appendices to Chapter 99.A.1. Weyl groups9.A.2. Spectral sequencesBibliographyIndex