Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions). - Hardcover

Helgason, Sigurdur

Hardcover

ISBN 10: 0821826735 ISBN 13: 9780821826737

Publisher: Amer Mathematical Society, 2000

This specific ISBN edition is currently not available.

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.

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From the Publisher

The author was awarded "The Steele Prize: of the American Mathematical Society in 1988 for this book together with its companion volume "Differential Geometry, Lie Groups and Symmetric Spaces," Academic Press, 1978.

Review

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. -- M. Flensted-Jensen, Bulletin of the London Mathematical Society, 1989

The book is excellent both as a text and as a reference work; it will clearly become another instant classic. -- A. Koranyi, American Scientist, 1985

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. --E.P. van den Ban, CWI Quarterly, 1987, p.35.

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