Helgason, Sigurdur Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. (English) Zbl 0965.43007 Mathematical Surveys and Monographs. 83. Providence, RI: American Mathematical Society (AMS). xxii, 667 p. (2000). This book is written by S. Helgason, an author of a number of excellent books in Lie theory and harmonic analysis on symmetric spaces which have already become classic. The book under review is a new printing of Helgason’s book “Groups and Geometric Analysis”, Pure Appl. Math., vol. 113 (Orlando etc. 1984; Zbl 0543.58001). It is devoted to some actual questions of harmonic analysis: integral geometry, invariant differential operators and spherical functions. It is both a textbook and a reference work on these topics. In the new printing the author corrected some inaccuracies and added a number of footnotes which are placed at the end of the book in a special section. Some of these footnotes furnish alternative, more elementary proofs and some are clarifications of occasional tough spots in the exposition. Finally, an errata list is added. Reviewer: V.F.Molchanov (Tambov) Cited in 1 ReviewCited in 174 Documents MSC: 43A85 Harmonic analysis on homogeneous spaces 43A90 Harmonic analysis and spherical functions 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53C65 Integral geometry 22E46 Semisimple Lie groups and their representations 53C35 Differential geometry of symmetric spaces 58C35 Integration on manifolds; measures on manifolds Keywords:Radon transform; Paley-Wiener theorem; Plancherel formula; symmetric spaces; harmonic analysis; integral geometry; invariant differential operators; spherical functions Citations:Zbl 0543.58001 PDFBibTeX XMLCite \textit{S. Helgason}, Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0965.43007)