eBook ISBN:  9781470413699 
Product Code:  SURV/142.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470413699 
Product Code:  SURV/142.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsMathematical Surveys and MonographsVolume: 142; 2007; 387 ppMSC: Primary 20; 22; 53
This book starts with the basic theory of topological groups, harmonic analysis, and unitary representations. It then concentrates on geometric structure, harmonic analysis, and unitary representation theory in commutative spaces. Those spaces form a simultaneous generalization of compact groups, locally compact abelian groups, and riemannian symmetric spaces. Their geometry and function theory is an increasingly active topic in mathematical research, and this book brings the reader up to the frontiers of that research area with the recent classifications of weakly symmetric spaces and of Gelfand pairs.
Part 1, “General Theory of Topological Groups”, is an introduction with many examples, including all of the standard semisimple linear Lie groups and the Heisenberg groups. It presents the construction of Haar measure, the invariant integral, the convolution product, and the Lebesgue spaces.
Part 2, “Representation Theory and Compact Groups”, provides background at a slightly higher level. Besides the basics, it contains the Mackey LittleGroup method and its application to Heisenberg groups, the Peter–Weyl Theorem, Cartan's highest weight theory, the Borel–Weil Theorem, and invariant function algebras.
Part 3, “Introduction to Commutative Spaces”, describes that area up to its recent resurgence. Spherical functions and associated unitary representations are developed and applied to harmonic analysis on \(G/K\) and to uncertainty principles.
Part 4, “Structure and Analysis for Commutative Spaces”, summarizes riemannian symmetric space theory as a rôle model, and with that orientation delves into recent research on commutative spaces. The results are explicit for spaces \(G/K\) of nilpotent or reductive type, and the recent structure and classification theory depends on those cases.
Parts 1 and 2 are accessible to firstyear graduate students. Part 3 takes a bit of analytic sophistication but generally is accessible to graduate students. Part 4 is intended for mathematicians beginning their research careers as well as mathematicians interested in seeing just how far one can go with this unified view of algebra, geometry, and analysis.
ReadershipGraduate students and research mathematicians interested in lie groups and their representations.

Table of Contents

Part 1. General theory of topological groups

1. Basic topological group theory

2. Some examples

3. Integration and convolution

Part 2. Representation theory and compact groups

4. Basic representation theory

5. Representations of compact groups

6. Compact Lie groups and homogeneous spaces

7. Discrete cocompact subgroups

Part 3. Introduction to commutative spaces

8. Basic theory of commutative spaces

9. Spherical transforms and Plancherel formulae

10. Special case: Commutative groups

Part 4. Structure and analysis for commutative spaces

11. Riemannian symmetric spaces

12. Weakly symmetric and reductive commutative spaces

13. Structure of commutative Nilmanifolds

14. Analysis on commutative Nilmanifolds

15. Classification of commutative spaces


Additional Material

Reviews

Wolf's book is an uptodate presentation of the harmonic analysis and classification theory of commutative spaces. He needs only 360 pages and amazingly few prerequisites to give complete proofs of all the results alluded to in this review.
Mathematical Reviews


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This book starts with the basic theory of topological groups, harmonic analysis, and unitary representations. It then concentrates on geometric structure, harmonic analysis, and unitary representation theory in commutative spaces. Those spaces form a simultaneous generalization of compact groups, locally compact abelian groups, and riemannian symmetric spaces. Their geometry and function theory is an increasingly active topic in mathematical research, and this book brings the reader up to the frontiers of that research area with the recent classifications of weakly symmetric spaces and of Gelfand pairs.
Part 1, “General Theory of Topological Groups”, is an introduction with many examples, including all of the standard semisimple linear Lie groups and the Heisenberg groups. It presents the construction of Haar measure, the invariant integral, the convolution product, and the Lebesgue spaces.
Part 2, “Representation Theory and Compact Groups”, provides background at a slightly higher level. Besides the basics, it contains the Mackey LittleGroup method and its application to Heisenberg groups, the Peter–Weyl Theorem, Cartan's highest weight theory, the Borel–Weil Theorem, and invariant function algebras.
Part 3, “Introduction to Commutative Spaces”, describes that area up to its recent resurgence. Spherical functions and associated unitary representations are developed and applied to harmonic analysis on \(G/K\) and to uncertainty principles.
Part 4, “Structure and Analysis for Commutative Spaces”, summarizes riemannian symmetric space theory as a rôle model, and with that orientation delves into recent research on commutative spaces. The results are explicit for spaces \(G/K\) of nilpotent or reductive type, and the recent structure and classification theory depends on those cases.
Parts 1 and 2 are accessible to firstyear graduate students. Part 3 takes a bit of analytic sophistication but generally is accessible to graduate students. Part 4 is intended for mathematicians beginning their research careers as well as mathematicians interested in seeing just how far one can go with this unified view of algebra, geometry, and analysis.
Graduate students and research mathematicians interested in lie groups and their representations.

Part 1. General theory of topological groups

1. Basic topological group theory

2. Some examples

3. Integration and convolution

Part 2. Representation theory and compact groups

4. Basic representation theory

5. Representations of compact groups

6. Compact Lie groups and homogeneous spaces

7. Discrete cocompact subgroups

Part 3. Introduction to commutative spaces

8. Basic theory of commutative spaces

9. Spherical transforms and Plancherel formulae

10. Special case: Commutative groups

Part 4. Structure and analysis for commutative spaces

11. Riemannian symmetric spaces

12. Weakly symmetric and reductive commutative spaces

13. Structure of commutative Nilmanifolds

14. Analysis on commutative Nilmanifolds

15. Classification of commutative spaces

Wolf's book is an uptodate presentation of the harmonic analysis and classification theory of commutative spaces. He needs only 360 pages and amazingly few prerequisites to give complete proofs of all the results alluded to in this review.
Mathematical Reviews