Geometric Analysis on Symmetric Spaces
The principal subject of this monograph is the analysis of Riemannian symmetric spaces of non-compact type. These are spaces of the form X = G/K, with G a real semisimple Lie group without compact factors; G is the group of isometries and K is the stabilizing subgroup of a fixed origin. The author’s 1978 monograph “Differential Geometry, Lie groups and Symmetric Spaces” gave an exposition of the basic geometric structure of these spaces and the connection with Lie theory. Helgason’s 1984 monograph “Groups and Geometric Analysis” dealt with the harmonic analysis of left K-invariant functions on X. The present monograph deals with the extension to harmonic analysis on X without assuming this left K-invariance. The theory is dominated by two integral transforms: on the one hand the Fourier transform, in terms of eigenfunctions on X for invariant differential operators, and on the other hand the horocycle transform. The latter transform is systematically viewed as a generalization of the classical Radon transform to the context of a so-called group invariant double fibration (an idea introduced by S.-S. Chern). The author systematically develops the theory of these transforms, building on Harish-Chandra’s spherical Plancherel formula and on the spherical Paley-Wiener theorem, two principal subjects of the 1984 monograph.
In the monograph, the general inversion and Paley-Wiener theorems for both the Fourier and Radon transform are discussed, as well as their connection and the relations with eigenspace representations and with the representations of the spherical principal series for G. Application to solvability of invariant differential equations is given. The treatment of all of these subjects relies on research contributions by the author throughout his career. The second edition differs only mildly from the first. The principal addition is the treatment of the multi-temporal wave equation, a remarkable hyperbolic system of equations first introduced by Semenov. The exposition, which emphasizes the geometric and analytic side of the subject, is self-contained and very clear. It is good that this standard in the field, with its wealth of material, has become available again through a second edition.