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Harmonic analysis of spherical functions on real reductive groups. (English) Zbl 0675.43004

Ergebnisse der Mathematik und ihrer Grenzgebiete, 101. Berlin etc.: Springer-Verlag. xiv, 365 p. (1988).
This book presents an extremely careful and well written account of Harish-Chandra’s theory of spherical functions and the related harmonic analysis on semisimple (or reductive) Lie groups. This is one of Harish- Chandra’s most beautiful contributions to non-commutative harmonic analysis.
It is the first textbook containing a complete treatment of the spherical Fourier transform (or the “Harish-Chandra”-transform as it is called in the book), including the transform on the Schwartz-spaces. (S. Helgason’s book [Groups and Geometric Analysis (Academic Press, 1984; Zbl 0543.58001)] does not treat the Schwartz-spaces in detail.)
Essentially all results from Harish-Chandra’s two famous papers from 1958 on spherical functions [Am. J. Math. 80, 241-310, 553-613 (1958; Zbl 0093.128)] are covered. The book avoids Harish-Chandra’s use of his results on discrete series in the proof of the inversion formula by taking into account the contributions due to Helgason, Gangolli and Rosenberg proving the inversion formula at the same time as the Paley- Wiener theorem.
The \(L^ 2\)-Schwartz-space is treated following Harish-Chandra’s method, using rather delicate estimates for the elementary spherical function obtained by the method of descent. The \(L^ p\)-Schwartz-spaces for \(0<p<2\) are treated following the paper by Trombi and Varadarajan using their refinement of Harish-Chandra’s method.
When writing the book the authors shared with many others the impression that to treat the spherical Schwartz-spaces “it appears that the complete Harish-Chandra apparatus will have to be used (asymptotic analysis, spectral theory, wave packets etc.)” (p. 299). However, in the summer of 1989 J. P. Anker has announced an elementary proof of the Schwartz-space results. His method of proof consists of a careful reduction to the Paley-Wiener theorem for \(C_ c^{\infty}(G/K)\). At first sight this means that a large part of the book under review is not needed to prove the main theorems. However, the method of Anker does not seem to give the detailed knowledge about the asymptotics of the elementary spherical functions obtained by Harish-Chandra’s method. These estimates are important in other contexts. Another point is that Harish- Chandra’s methods are applicable in a much more general situation than that of K-biinvariant function on G. From this point of view the book is a most valuable introduction to Harish-Chandra’s methods in general.
With regards to the bibliography it is to my taste too short. Harish- Chandra’s theory of spherical functions has inspired much research related to the spherical Fourier transform on symmetric spaces, much more than is apparent from looking at the bibliography. The book refers to Helgason’s books for further references and this may compensate somewhat. In particular, I miss a reference to J.-L. Clerc [J. Funct. Anal. 37, 182-202 (1980; Zbl 0507.43002)]. In this paper Clerc gives a proof of the \(L^ p\)-Schwartz-theorem by reducing it to the \(L^ 2\)-Schwartz- theorem.
Reviewer: M.Flensted-Jensen

MSC:

43A90 Harmonic analysis and spherical functions
22E30 Analysis on real and complex Lie groups
43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
22-02 Research exposition (monographs, survey articles) pertaining to topological groups