Oxford: Clarendon Press. xii, 382 p. £ 50.00 (1994).
Two most interesting examples of symmetric cones are the cone of real positive definite symmetric matrices and the Lorentz cone. Harmonic analysis on the cone of real positive definite symmetric matrices has been developed in the past to a very high degree and this cone actually plays a fundamental role in two different fields: number theory and statistics. The Lorentz cone has been studied in depth, mainly from the point of view of the wave equation. But the general case is much less developed, even though basic work has been done and some important results have been proved by Gindikin and Vinberg.
The purpose of this book is to give a systematical exposition of the geometry of symmetric cones, and to develop analysis on these cones and on the complex tube domains associated with them. The approach is based on the theory of Jordan algebras in the way worked out by M. Koecher and his school. This approach gives explicit formulas and also a classification of the symmetric cones relatively easily. Symmetric cones and tubes over them are examples of Riemannian symmetric spaces. The class of symmetric cones has special features which make it possible to go much further in explicit harmonic analysis than in the general case of arbitrary Riemannian symmetric spaces. This is due to the interplay of Euclidean harmonic analysis on the ambient vector space, and of the non-commutative harmonic analysis of the cone as a symmetric space.
The contents of the book are the following: In the first chapter symmetric cones are introduced, and in the second chapter the basic properties of Jordan algebras are presented. With the aid of these in Chapter III the fundamental result due to M. Koecher and E. B. Vinberg can be stated that the interior of the cone of squares in a Euclidean Jordan algebra is a symmetric cone, and every symmetric cone is obtained in this way. In Chapter IV the Peirce decomposition is presented. This gives an algebraic technique making it possible to regard a Jordan algebra as a generalization of the space of symmetric matrices. It allows us to give a classification of Euclidean Jordan algebras and symmetric cones in Chapter V. In Chapter VI generalized versions of the diagonalization and triangularization of matrices are discussed. The interplay of Euclidean harmonic analysis and of non-commutative harmonic analysis gives rise to the theory of the gamma function of a symmetric cone which is introduced in Chapter VII. It plays a central role in this book. In Chapter VIII it is proved that any semisimple Jordan algebra is the complexification of a Euclidean one.
In Chapter IX complex tube domains over convex cones are studied and the discussion of the Bergman and Hardy spaces on the tube domain is given. In Chapter X tube domains over symmetric cones are specialized and it allows us to obtain explicit formulas for the Bergman and Cauchy-Szegö kernels involving the determinant function of the underlying Jordan algebra. Another point where Euclidean analysis and non-commutative harmonic analysis come together is the theory of spherical polynomials, introduced in Chapter XI. They can be used to define generalized Taylor and Laurent expansions which are studied in Chapter XII.
In Chapter XIII the function spaces on symmetric domains of tube type are studied and the Wallach set is determined. In Chapter XIV it is shown that the spherical functions of a symmetric cone, considered as a Riemannian symmetric space, and the spherical Fourier transform, can be written explicitly in terms of the Jordan algebra structure. The gamma function of a symmetric cone leads naturally to generalized hypergeometric expansions which are studied in Chapter XV, with particular emphasis on the Bessel functions and the Gauss hypergeometric functions. In Chapter XVI it is shown that in some cases it is possible to consider a symmetric cone as a set of dilations of a representation space of the Jordan algebra and to obtain in that way analogs of some classical objects such as the Hankel transform.