[...].

Symmetric spaces are hugely important objects, occurring in many different parts of contemporary mathematics. In number theory, for example, one cannot even begin to do modular forms seriously without a reasonably thorough knowledge of the symmetric space afforded by the action of SL(2,Z) on the complex upper half plane. The indicated discrete group acts on {Im(z) 0} by fractional linear transformations resulting in a canonical simply connected orbit space, the fundamental domain, which, upon one-point compactification, acquires the structure of a compact Riemann surface. This beautiful and rich special case becomes the exemplar for the general state of affairs as far as automorphic functions are concerned: the moment a discrete (matrix) group acts on an appropriate space, typically situated in some Cr, the orbit space can be finessed so as to bring in a lot of exceedingly useful geometry and analysis. Voilá: a symmetric space.
Furthermore, aside from their natural appearance in nature, symmetric spaces are exceedingly interesting things in themselves, and have over the years given rise to a vast amount of literature. At the very top of this list one encounters the indicated contributions by Sigurdur Helgason, one of the true grandmasters of the genre. The book under review, Geometric Analysis on Symmetric Spaces, published by the AMS, is the second edition of the 1994 original, and is one of a quartette of seminal texts by this author, the other three being Differential Geometry, Lie Groups, and Symmetric Spaces (AMS, 2001), Groups and Geometric Analysis (AMS, 2000) --- these two books earned Helgason the Steele Prize ---, and, of course, the 1962 classic Differential Geometry and Symmetric Spaces. Manifestly, when it comes to symmetric spaces, Helgason's books are a sine qua non.
This having been said, Geometric Analysis on Symmetric Spaces is intended to be "easily accessible to readers with some modest background in Lie group theory which by now is widely known ..." (fair enough) and focuses "on analysis on Riemannian symmetric spaces X = G/K." Helgason addresses, among other things, "existence and uniqueness theorems for invariant differential equations on X, explicit solution formulas, as well as geometric properties of the solutions, for example the harmonic functions and the wave equation on X." The book relies to some degree on Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis, but autonomy is striven for, as is accessibility: "To facilitate self-study and to indicate further developments each chapter concludes with a section `Exercises and Further Results.'"
At over 600 pages this is no quick jaunt in the park: it is serious mathematical exploration and the reader should be prepared for this sort of austerity. However, it is ultimately a foregone conclusion that Geometric Analysis and Symmetric Spaces is a model of fine scholarship and must rank as a definitive source for the indicated material. From my own parochial point of view, Chapter III, "The Fourier Transform on a Symmetric Space," and Chapter VI, "Eigenspace Representations," hold the most promise of all, but it is all most exciting and well worth the effort.

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