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Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions
Sigurdur Helgason, Massachusetts Institute of Technology, Cambridge, MA
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Mathematical Surveys and Monographs
1984; 667 pp; hardcover
Volume: 83
Reprint/Revision History:
reprinted 2002
ISBN-10: 0-8218-2673-5
ISBN-13: 978-0-8218-2673-7
List Price: US$65
Member Price: US$52
Order Code: SURV/83
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This volume, the second of Helgason's impressive three books on Lie groups and the geometry and analysis of symmetric spaces, is an introduction to group-theoretic methods in analysis on spaces with a group action.

The first chapter deals with the three two-dimensional spaces of constant curvature, requiring only elementary methods and no Lie theory. It is remarkably accessible and would be suitable for a first-year graduate course. The remainder of the book covers more advanced topics, including the work of Harish-Chandra and others, but especially that of Helgason himself. Indeed, the exposition can be seen as an account of the author's tremendous contributions to the subject.

Chapter I deals with modern integral geometry and Radon transforms. The second chapter examines the interconnection between Lie groups and differential operators. Chapter IV develops the theory of spherical functions on semisimple Lie groups with a certain degree of completeness, including a study of Harish-Chandra's \(c\)-function. The treatment of analysis on compact symmetric spaces (Chapter V) includes some finite-dimensional representation theory for compact Lie groups and Fourier analysis on compact groups. Each chapter ends with exercises (with solutions given at the end of the book!) and historical notes.

This book, which is new to the AMS publishing program, is an excellent example of the author's well-known clear and careful writing style. It has become the standard text for the study of spherical functions and invariant differential operators on symmetric spaces.

Sigurdur Helgason was awarded the Steele Prize for Groups and Geometric Analysis and the companion volume, Differential Geometry, Lie Groups and Symmetric Spaces.

Readership

Graduate students and research mathematicians interested in analysis on homogeneous spaces, differential geometry, and topological groups, Lie groups.

Reviews

From reviews of the original edition ...

"The book is excellent both as a text and as a reference work; it will clearly become another instant classic."

-- American Scientist

"This volume makes an excellent companion to the author's Differential Geometry, Lie Groups, and Symmetric Spaces, putting to work many of the abstract concepts developed in the earlier volume. The introductory material and large number of exercises (with answers!) will make the book quite appropriate for students. Researchers will find numerous useful references on geometric analysis, along with proofs, connections with other parts of mathematics, and valuable historical remarks.

"This book, like the author's previous work on differential geometry, will no doubt inspire considerable further research and become the standard text on the subjects it covers."

-- Mathematical Reviews

"Few treatises today can lay claim to being "aere perennius", but all of Helgason's books certainly do with a vengeance ... [He] sets a model of style and clarity that has not been matched since Enriques's Geometria proiettiva. This is the kind of mathematics that will live forever."

-- The Bulletin of Mathematics Books

"A most valuable contribution to Lie theory and to the interplay between geometry and analysis. It is remarkable that the beautiful theory in Chapter IV can be presented in a textbook form with complete proofs."

-- Bulletin of the London Mathematical Society

"The diversity of subjects treated is great. Nevertheless the author has managed to achieve coherence of presentation by clearly putting forward a few main themes and basic problems. The first third of the book is suitable as a text for beginning graduate students; the book is also an excellent source of reference for experts. No doubt it will become a new standard in the field."

-- CWI Quarterly

Table of Contents

  • Geometric Fourier analysis on spaces of constant curvature
  • Integral geometry and Radon transforms
  • Invariant differential operators
  • Invariants and harmonic polynomials
  • Spherical functions and spherical transforms
  • Analysis on compact symmetric spaces
  • Appendix
  • Some details
  • Bibliography
  • Symbols frequently used
  • Index
  • Errata
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