##
**Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions.**
*(English)*
Zbl 0543.58001

Since the author’s well-known book (1) Differential geometry and symmetric spaces (1962; Zbl 0111.181), analysis on Lie groups and symmetric spaces has developed considerably, and called for an updated and expanded version of (1). This was taken up by (2) Differential geometry, Lie groups, and symmetric spaces (1978; Zbl 0451.53038) first, which covered the basic theory of symmetric spaces and semisimple Lie groups, culminating in E. Cartan’s classification. Analysis on these objects begins with the present book, which can be considered as a greatly expanded version of Chapter X in (1).

The title lists the topics exposed here. Not included are solvability problems for differential operators, eigenspace representations and nonspherical harmonic analysis on a noncompact symmetric space, left over for a forthcoming book.

The 80 page introduction deals with the Euclidean plane \({\mathbb{R}}^ 2\), the sphere \(S^ 2\), and the hyperbolic plane \(H^ 2\). In these cases, elementary and explicit calculations provide complete solutions to the main problems in Fourier analysis. With emphasis on the role played by the corresponding groups of isometries, and algebras of invariant differential operators (i.e. the Laplace operator, here), this preliminary chapter gives the flavour of the whole book, and an easy introduction to spherical functions, c-functions, eigenfunctions of the Laplacian, eigenspace representations....

Chapter I, ”Integral geometry and Radon transforms” (153 pages), contains the basic results about invariant measures on homogeneous spaces of Lie groups, and the classical integral formulas for semisimple Lie groups. The Radon transform is then studied on \({\mathbb{R}}^ n\); it associates to a function the family of its integrals over all hyperplanes. Support theorems and inversion formulas are proved, with applications to the wave equation. This example motivates the general definition of Radon transforms for homogeneous spaces in duality \(({\mathbb{R}}^ n\) and \({\mathbb{P}}^ n\) above); a detailed study of this transform is given for compact and noncompact two-point homogeneous spaces, with explicit inversion formulas. The chapter ends with a solution of the orbital integral problem (find a function, knowing its integrals over generalized spheres) for the isotropic Lorentzian manifolds and for \(G\times G/diagonal\), \(G=SL(2,{\mathbb{R}})\); the latter case is an essential step in the proof of the Plancherel formula for G.

Chapter II, ”Invariant differential operators” (112 pages), begins with general properties of differential operators on manifolds (including Peetre’s characterization). The author’s geometric theory of differential operators (projection, transversal and radial part) follows, with explicit results for the Laplacian and detailed study of several examples in the context of semisimple Lie groups. The last sections of the chapter contain many results on the algebra of invariant differential operators on general homogeneous spaces, and symmetric spaces in particular. Classical mean value theorems related to the Laplacian are extended to rank one symmetric spaces.

Chapter III, ”Invariants and harmonic polynomials” (40 pages), contains Chevalley’s theorem on invariant polynomials, structure theorems on the symmetric (or exterior) algebra invariants under a linear group action, on the space of harmonic polynomials, and Kostant-Rallis’ results on the orbit structure in the tangent space of a symmetric space.

Chapter IV, ”Spherical functions and spherical transforms” (110 pages), is perhaps the richest of the book. After some general results concerning spherical functions on Riemannian homogeneous spaces, and their link to representation theory, the author specializes to symmetric spaces of the noncompact type G/K, G semisimple, K maximal compact. The now classical Harish-Chandra expansion of spherical functions is given, as well as a complete proof of Gindikin-Karpelevic’s formula for the c-function, of the inversion formula and Paley-Wiener theorem for the spherical transform, and Helgason-Johnson’s characterization of bounded spherical functions. For G complex, the theory gets much simpler. The chapter ends with spherical analysis on the tangent space to G/K, and with Kostant’s convexity theorems.

Chapter V, ”Analysis on compact symmetric spaces” (56 pages), contains the representation theory of simply connected compact semisimple Lie groups, including Kirillov’s character formula for this case. In the next section, several classical results on Fourier series on the circle appear linked together when extended to arbitrary compact groups. The book ends with a study of harmonic analysis, eigenfunctions, and eigenspace representations for a symmetric space of the compact type.

The material in this book comes from papers by Harish-Chandra, Kostant, and others, but mostly from the work of the author himself, thus initiating a fine synthesis of his results. The book is self-contained, to a large extent. 129 exercises, with solutions or references to the literature, provide additional examples and further results, such as Duflo’s isomorphism, Flensted-Jensen’s method of duality, Kostant’s multiplicity formula.... Detailed historical notes, a 35-page bibliography, and an appendix complete the book. Independence of chapters, intentional repetitions (e.g. some particular study before the general case), a substantial alphabetical index, and the author’s clear style, make it a very accessible textbook.

The title lists the topics exposed here. Not included are solvability problems for differential operators, eigenspace representations and nonspherical harmonic analysis on a noncompact symmetric space, left over for a forthcoming book.

The 80 page introduction deals with the Euclidean plane \({\mathbb{R}}^ 2\), the sphere \(S^ 2\), and the hyperbolic plane \(H^ 2\). In these cases, elementary and explicit calculations provide complete solutions to the main problems in Fourier analysis. With emphasis on the role played by the corresponding groups of isometries, and algebras of invariant differential operators (i.e. the Laplace operator, here), this preliminary chapter gives the flavour of the whole book, and an easy introduction to spherical functions, c-functions, eigenfunctions of the Laplacian, eigenspace representations....

Chapter I, ”Integral geometry and Radon transforms” (153 pages), contains the basic results about invariant measures on homogeneous spaces of Lie groups, and the classical integral formulas for semisimple Lie groups. The Radon transform is then studied on \({\mathbb{R}}^ n\); it associates to a function the family of its integrals over all hyperplanes. Support theorems and inversion formulas are proved, with applications to the wave equation. This example motivates the general definition of Radon transforms for homogeneous spaces in duality \(({\mathbb{R}}^ n\) and \({\mathbb{P}}^ n\) above); a detailed study of this transform is given for compact and noncompact two-point homogeneous spaces, with explicit inversion formulas. The chapter ends with a solution of the orbital integral problem (find a function, knowing its integrals over generalized spheres) for the isotropic Lorentzian manifolds and for \(G\times G/diagonal\), \(G=SL(2,{\mathbb{R}})\); the latter case is an essential step in the proof of the Plancherel formula for G.

Chapter II, ”Invariant differential operators” (112 pages), begins with general properties of differential operators on manifolds (including Peetre’s characterization). The author’s geometric theory of differential operators (projection, transversal and radial part) follows, with explicit results for the Laplacian and detailed study of several examples in the context of semisimple Lie groups. The last sections of the chapter contain many results on the algebra of invariant differential operators on general homogeneous spaces, and symmetric spaces in particular. Classical mean value theorems related to the Laplacian are extended to rank one symmetric spaces.

Chapter III, ”Invariants and harmonic polynomials” (40 pages), contains Chevalley’s theorem on invariant polynomials, structure theorems on the symmetric (or exterior) algebra invariants under a linear group action, on the space of harmonic polynomials, and Kostant-Rallis’ results on the orbit structure in the tangent space of a symmetric space.

Chapter IV, ”Spherical functions and spherical transforms” (110 pages), is perhaps the richest of the book. After some general results concerning spherical functions on Riemannian homogeneous spaces, and their link to representation theory, the author specializes to symmetric spaces of the noncompact type G/K, G semisimple, K maximal compact. The now classical Harish-Chandra expansion of spherical functions is given, as well as a complete proof of Gindikin-Karpelevic’s formula for the c-function, of the inversion formula and Paley-Wiener theorem for the spherical transform, and Helgason-Johnson’s characterization of bounded spherical functions. For G complex, the theory gets much simpler. The chapter ends with spherical analysis on the tangent space to G/K, and with Kostant’s convexity theorems.

Chapter V, ”Analysis on compact symmetric spaces” (56 pages), contains the representation theory of simply connected compact semisimple Lie groups, including Kirillov’s character formula for this case. In the next section, several classical results on Fourier series on the circle appear linked together when extended to arbitrary compact groups. The book ends with a study of harmonic analysis, eigenfunctions, and eigenspace representations for a symmetric space of the compact type.

The material in this book comes from papers by Harish-Chandra, Kostant, and others, but mostly from the work of the author himself, thus initiating a fine synthesis of his results. The book is self-contained, to a large extent. 129 exercises, with solutions or references to the literature, provide additional examples and further results, such as Duflo’s isomorphism, Flensted-Jensen’s method of duality, Kostant’s multiplicity formula.... Detailed historical notes, a 35-page bibliography, and an appendix complete the book. Independence of chapters, intentional repetitions (e.g. some particular study before the general case), a substantial alphabetical index, and the author’s clear style, make it a very accessible textbook.

Reviewer: F.Rouvière

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

53C65 | Integral geometry |

43A90 | Harmonic analysis and spherical functions |

43A85 | Harmonic analysis on homogeneous spaces |

22E30 | Analysis on real and complex Lie groups |