
Harmonic Analysis on Commutative Spaces
Joseph A. Wolf(Author)
American Mathematical Society (Publisher)
Published on 1. August 2007
Book
Hardback
392 pages
978-0-8218-4289-8 (ISBN)
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Description
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 Little-Group 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 role 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 first-year 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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
College/higher education
Illustrations
Illustrations
ISBN-13
978-0-8218-4289-8 (9780821842898)
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Schweitzer Classification
Content
General theory of topological groups: Basic topological group theory Some examples Integration and convolution Representation theory and compact groups: Basic representation theory Representations of compact groups Compact Lie groups and homogeneous spaces Discrete co-compact subgroups Introduction to commutative spaces: Basic theory of commutative spaces Spherical transforms and Plancherel formulae Special case: Commutative groups Structure and analysis for commutative spaces: Riemannian symmetric spaces Weakly symmetric and reductive commutative spaces Structure of commutative nilmanifolds Analysis on commutative nilmanifolds Classification of commutative spaces Bibliography Subject index Symbol index Table index.