The study of causal symmetric spaces originated in the early 1980s with the work of G. I. Ol’shanskij on the realization of the holomorphic discrete series of a Hermitian group on a Hardy space of holomorphic functions on a subsemigroup of its complexification [cf., e.g., Vopr. Teor. Grupp Gomologicheskoj Algebry 1982, 85-98 (1982; Zbl 0581.22012)], work that embodied aspects of the Gelfand-Gindikin program. Fueled by cross-connections with the Lie theory of semigroups and the theory of ordered homogeneous spaces, and by applications to harmonic analysis and representation theory, the theory has experienced explosive growth since those early beginnings.
The book under review provides the first coherent and unified introduction to the theory of causal symmetric spaces. It develops the structure and geometry of causal symmetric spaces, primarily for irreducible semisimple symmetric spaces, in the first six chapters and turns to their associated harmonic analysis in the last three chapters.
A causal structure on a manifold \(M\) is an assignment of a closed convex cone \(\Theta(x)\) (modeled after the light cone in relativity) in the tangent space of each point \(x\). Such a structure gives rise to an “order” on \(M\) by saying that \(x\) precedes \(y\) if one can travel from \(x\) to \(y\) along a curve whose derivative lies in the assigned cone at each point. If the manifold \(M\) is a homogeneous space with respect to the action of a Lie group \(G\), then it is natural to require the field of cones to be invariant, i.e., \(g\cdot\Theta(x)=\Theta(g\cdot x)\) holds for all \(g\) and \(x\). Such an invariant causal structure is called a causal orientation. In this case, a single cone in the tangent space of some fixed point \({\mathbf o}\) completely determines the invariant field of cones. The set of points preceded by \({\mathbf o}\) may be viewed as a positive domain in \(M\), and the set of group elements mapping the positive domain into itself is a semigroup, called the causal semigroup, which is one effective tool (at the group level) in the study of causal orientations. Precisely those cones in the tangent space at \({\mathbf o}\) that are invariant under the action of the stabilizer group \(H\) of \({\mathbf o}\) give rise to causal orientations. Suppose additionally that \(M=G/H\) is a symmetric space. Then, there is an involution \(\tau:G\to G\) with \(H\) an open subgroup of the fixed-point group \(G^\tau\) whose infinitesimal version \(d\tau\) yields an eigenspace decomposition \({\mathfrak g}={\mathfrak h}+{\mathfrak q}\) of the Lie algebra \({\mathfrak g}\) of \(G\), where \({\mathfrak h}\), the Lie algebra of \(H\), is the eigenspace for \(+1\) and \({\mathfrak q}\) is the eigenspace for \(-1\). The tangent space of \(M\) at \({\mathbf o}\) can be identified with \({\mathfrak q}\), and the causal orientations are in bijective correspondence with the \(H\)-invariant cones in \({\mathfrak q}\). It is always assumed that the cones are regular, i.e., closed, convex, \({\mathfrak q}\)-generating, and pointed. A symmetric space equipped with a causal orientation is a causal symmetric space, the subject of this book, and their study via Lie algebra methods in the \(H\)-module \({\mathfrak q}\) is an absolutely fundamental tool of the subject.
The first two chapters introduce the basic concepts concerning causality, associated semigroups, and convex cones, and discuss fundamental examples. If there is an \(H\)-invariant cone \(C\subseteq{\mathfrak q}\) whose interior consists of elliptic (hyperbolic) elements, then \(M\) is said to be (non-)compactly causal and is said to have Cayley type if it is both compactly and non-compactly causal. Under the basic duality \({\mathfrak g}\to {\mathfrak g}^c:={\mathfrak h}+i{\mathfrak q}\), non-compactly causal symmetric spaces correspond to compactly causal ones and vice-versa. This duality reduces the classification of irreducible causal symmetric spaces to the case of compactly causal spaces. The latter are classified via their bijective correspondence with antiholomorphic involutions of bounded symmetric domains (Chapter 3).
In Chapter 4, the cones in \({\mathfrak q}\) leading to causal orientations on the previously described symmetric spaces are determined; duality insures that it suffices to consider the non-compact case. The cones are uniquely determined by their intersection with a Cartan-like abelian subspace \({\mathfrak a}\) of \({\mathfrak q}\), and it is possible to characterize the cones that occur as intersections with \({\mathfrak a}\) as those sandwiched between certain cones \(c_{max}\) and \(c_{min}\) and invariant under the Weyl group. An important tool in carrying out this program is the linear convexity theorem of Ólafsson.
The order of precedence on a noncompactly causal symmetric space is a closed partial order which is globally hyperbolic, that is, the order intervals are compact (Chapter 5). Moreover, the causal semigroup associated to the maximal causal structure is characterized as the compression semigroup leaving a certain open domain in a flag manifold of \(G\) invariant. The causal semigroup admits an Iwasawa-like decomposition which makes it possible to describe the positive cone of the symmetric space purely in terms of the “solvable part” of the semigroup. These are valuable tools in the study of the order compactification of a noncompactly causal symmetric space and yield an explicit picture of the \(G\)-orbit structure of the order compactification (Chapter 6).
The authors impressively demonstrate in the last three chapters how harmonic analysis and operator theory on causal symmetric spaces can be built up from their rich geometric structure. These chapters are more expository in nature and contain discussions and complete statements of results as well as explicit calculations for low-dimensional examples, but hardly any proofs. The authors first return to the roots of the subject and discuss representations of \(G\) in the context of compactly causal symmetric spaces (Chapter 7). The representations considered are the so-called highest weight unitary representations, which can be characterized as those admitting an extension to a holomorphic representation of a certain complex semigroup. This leads to the construction of Hardy spaces for causal symmetric spaces and a description of the decomposition of the Hardy space under the regular action of \(G\). For noncompactly causal symmetric spaces, the authors consider the causal kernels on the space \(M\), continuous functions on \(M\times M\) supported on the order \(\{(x,y):x\leq y\}\) (Chapter 8). The invariant kernels form a commutative algebra with respect to kernel multiplication and this algebra is studied via the spherical Laplace transform. This leads to the introduction of spherical functions, the study of their asymptotic behavior, and an inversion formula. Many of these ideas generalize classical notions carried over from the theory of Riemannian symmetric spaces, but one must typically do extra work to compensate for the non-compactness of \(H\). There is, associated with a non-compactly causal symmetric space \(M\), an algebra of Wiener-Hopf operators that arises as an algebra of operators on the \(L^2\)-space of the future of a point in \(M\). Information about the causal compactification of \(M\) can be used to describe the ideal structure of this algebra (Chapter 9).
Besides being an excellent introduction to the structure and geometry of causal symmetric spaces and a convenient and accessible reference source, this book (particularly the latter chapters) can be used as a guide to more recent developments in the area and a lead-in to current open problems and active areas of research.