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Harmonic analysis on commutative spaces. (English) Zbl 1156.22010
Mathematical Surveys and Monographs 142. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4289-8). xv, 387 p. (2007).
The subject of this book are Gelfand pairs (or ‘commutative pairs’) $$(G,K)$$ and the harmonic analysis on the associated homogeneous spaces $$G/K$$ (the ‘commutative spaces’). Thus $$G$$ is a locally compact topological group and $$K$$ a compact subgroup of $$G$$; such a pair $$(G,K)$$ is called commutative if $$L^1(K\backslash G/K)$$ (the convolution algebra of bi-$$K$$-invariant elements of $$L^1(G)$$) is commutative. Harmonic analysis on commutative spaces is a common roof for various classical areas of harmonic analysis, like harmonic analysis on a locally compact abelian group $$(G,+)$$ (take $$K:=\{0\}$$) or on a compact group $$H$$ (take $$G:=H\times H$$ and let $$K$$ be the diagonal in $$G$$). Further prime examples of commutative pairs $$(G,K)$$ arise from Riemannian symmetric spaces $$X=G/K$$, when $$G$$ is the largest connected group of isometries of $$X$$.
The book under review provides an introduction to harmonic analysis on commutative spaces, leading from the foundations to recent research. It is structured into four parts of increasing sophistication:
Part 1. General theory of topological groups.
Part 2. Representation theory and compact groups.
Part 3. Introduction to commutative spaces.
Part 4. Structure and analysis for commutative spaces.
The first three parts are introductory, and have been the subject of courses taught by the author. Parts 1 and 2 prepare the reader for the main theme of the book. Part 3 introduces the core concepts. The final part is devoted to more advanced topics, like classification problems.
The book begins with an introduction to topological groups, written with a view towards the main topic. After some basic definitions and results, assorted examples (mostly linear Lie groups) are given for later use. The high point of Part 1 is a discussion of Haar measure on locally compact groups, together with the associated function spaces and convolution algebras. The prerequisites for reading this part are moderate: It assumes a working knowledge of point-set topology, rudiments of algebraic topology (fundamental group, covering spaces) and basics of measure theory on locally compact spaces (Riesz representation theorem, products of Radon measures).
Part 2 starts with an introduction to the representation theory of locally compact groups (Chapter 4). Besides unitary representations on Hilbert spaces, also uniformly bounded representations on Banach spaces are discussed to some extent. After basic topics like the construction of new representations from given ones (via subrepresentations, quotient representations, direct sums and tensor products), multiplicity-free representations are characterized using the commuting algebra. Moreover, unitary representations which are completely continuous (i.e., which induce a representation of the group algebra by compact operators) are shown to be finite-multiplicity discrete direct sums of irreducibles. Then direct integrals of unitary representations and induced representations are introduced. Moreover, Mackey’s Little Group Theorem is stated (without proof, and suppressing the measure-theoretic background), and applied to Heisenberg groups.
The next chapter is devoted to the representation theory of compact groups. All important standard facts (like the Peter-Weyl Theorem, the Plancherel Formula and the Frobenius Reciprocity Theorem) are established.
The demands on the reader increase in Chapter 6, which explains how the theory specializes if the compact group under consideration is a Lie group. While differential geometry and Lie theory could be avoided almost entirely in earlier chapters, this stops to be possible at this point. To assist the reader, a quite condensed ‘crash course’ on reductive Lie groups and Cartan’s highest weight theory is provided (without proofs). The chapter also contains a proof of the Borel-Weil Theorem and states the Borel-Weil-Bott Theorem.
Part 2 ends with a brief chapter on discrete co-compact subgroups $$\Gamma$$ of a locally compact group $$G$$ and representations in function spaces on the compact quotients $$G/\Gamma$$ (culminating in Selberg’s trace formula).
The discussion of commutative spaces begins in Part 3. The results include equivalent characterizations of the commutativity of a pair $$(G,K)$$ of a locally compact group $$G$$ and a compact subgroup $$K$$. For example, $$(G,K)$$ is commutative if and only if $$C_c(K\backslash G/ K)$$ is commutative under convolution, if and only if $$KgKg'K=Kg'KgK$$ for all $$g,g'\in G$$, if and only if the action of $$G$$ on $$L^2(G/K)$$ is multiplicity free. And if $$G$$ is a connected Lie group, then $$(G,K)$$ is commutative if and only if the algebra $${\mathcal D}(G,K)$$ of $$G$$-invariant differential operators on $$G/K$$ is commutative (as first shown by E.G.F. Thomas). In addition, some convenient criteria are provided. For instance, $$(G,K)$$ is commutative if $$G$$ admits an involutive automorphism $$\theta$$ such that $$\theta(g^{-1})\in KgK$$ for all $$g\in G$$ (as shown by Gelfand). This enables Riemannian symmetric spaces to be recognized as commutative spaces (as mentioned above). Another criterion assumes that $$X:=G/K$$ can be given a $$G$$-invariant distance function such that $$G$$ acts transitively on pairs of points in $$X$$ of each fixed distance. It applies if $$G$$ is the automorphism group of a homogeneous tree and $$K$$ the stabilizer of a vertex.
As concerns harmonic analysis, spherical functions for $$(G,K)$$ are introduced as continuous functions $$\omega\colon G\to{\mathbb C}$$ such that the measure $$dm=\omega(x^{-1})\,dx$$ on $$G$$ (with Haar measure $$dx$$) is bi-$$K$$-invariant and $$C_c(K\backslash G/K)\to{\mathbb C}$$, $$f\mapsto \int_G f\,dm$$ is an algebra homomorphism. The relations between positive definite spherical functions and (suitable) irreducible unitary representations of $$G$$ are explained, as are the fundamentals of harmonic analysis on $$G/K$$ going along with spherical functions. In particular, the spherical transform and its inverse are discussed, Bochner’s theorem, and the Plancherel theorem. Also uncertainty principles are broached. The final chapter of Part 3 explains how the general theory specializes in the case of locally compact abelian groups. The familiar facts are recovered (including the Pontryagin duality theorem).
The final part of the book begins with a chapter on Riemannian symmetric spaces (Chapter 11). In this case, the general theory can be made much more concrete; e.g., the spherical functions are reasonably explicit, and the Plancherel measure is completely explicit. It is an active area of research to find similar structural and analytical results for more general classes of commutative spaces, and the later chapters contain some recent results in this direction.
After a ‘crash course’ in Riemannian symmetric spaces and some classification results, the theory of spherical functions is worked out for euclidean spaces (considered as symmetric spaces in various ways) and for symmetric spaces of compact type. In the case of symmetric spaces of non-compact type, some facts on spherical functions and Plancherel measure are summarized.
A pair $$(G,K)$$ of a Lie group $$G$$ with finitely many connected components and a compact subgroup $$K$$ is called weakly symmetric if there exists an automorphism $$\sigma$$ of $$G$$ such that $$\sigma(g)\in Kg^{-1}K$$ for all $$g\in G$$. Chapter 12 is devoted to pairs $$(G,K)$$ with $$G$$ a reductive Lie group. Among other things, it is shown that $$(G,K)$$ is commutative if and only if it is weakly symmetric. The chapter culminates in O. Yakimova’s classification of the irreducible reductive weakly symmetric pairs [Gelfand pairs. Bonner Mathematische Schriften 374 (2005; Zbl 1073.22006)].
Chapter 13 deals with the structure of commutative pairs $$(G,K)$$ in which $$G$$ is a Lie group and a nilpotent subgroup $$N$$ of $$G$$ acts transitively on $$G/K$$. The harmonic analysis on the corresponding commutative nilmanifolds is the content of Chapter 14.
The final chapter summarizes the last two chapters of Yakimova’s thesis (cited above). Besides further classification results, a commutativity criterion is described for pairs $$(G,K)$$ in which $$G$$ is a semidirect product of its nilradical and a reductive group containing $$K$$.
It should be plain from the preceding overview that this book contains an impressive amount of important results. The presentation is clear and emphasizes the main ideas, which helps the reader to absorb the material (although some proofs are quite condensed). As already explained, the book is not entirely self-contained. But the author mitigates this problem by providing ‘crash courses’ on coherent portions of outside material (which are also of value to more advanced readers, as welcome reminders). It is a drawback that references to the literature are usually given globally (without indication of a page, section, or theorem); less experienced readers would probably be grateful for more guidance in this respect.
The reviewer found only few misprints and slips of the pen (e.g., Lemma 1.7.4 is wrong as $$G:={\mathbb Z}+{\mathbb Z}\sqrt{2}\subseteq {\mathbb R}$$ shows; continuity of $$\phi$$ in Proposition 8.4.2 (iv) is not a conclusion, but should be made a hypothesis). Two statements may be confusing for less experienced readers (but remain without consequence, and do not at all affect the qualities of the book):
I. As first shown by J. Dieudonné [C. R. Acad. Sci., Paris 218, 774–776 (1944; Zbl 0061.04205)], there are metrizable topological groups which do not admit completions, because they contain left Cauchy sequences which are not right Cauchy sequences (whence Theorem 1.8.5 and Lemma 1.8.3 are not correct).
II. Contrary to an assertion on page 67, the (completed) projective tensor product $$H_1\widehat{\otimes} H_2$$ of two Hilbert spaces $$H_1$$ and $$H_2$$ does not coincide with the Hilbert space tensor product $$H_1\widehat{\otimes}_H H_2$$ in general, as the example $$H_1:=H_2:=\ell^2$$ shows. In fact, the continuous bilinear map $$\ell^2 \times \ell^2\to \ell^1$$ $$(f,g)\mapsto (f(n)g(n))_{n\in {\mathbb N}}$$ cannot induce a continuous linear map $$\ell^2 \widehat{\otimes}_H \ell^2\to \ell^1$$, because the latter would take $$\sum_{n=1}^\infty \frac{1}{n}e_n\otimes e_n$$ to the non-summable sequence $$(\frac{1}{n})_{n\in {\mathbb N}}$$ (where $$e_n(m):=\delta_{n,m}$$). Using that $$H_j$$ is a complemented subspace of $$B(H_j)$$, a similar argument also shows that $$B(H_1\widehat{\otimes}_H H_2)$$ need not coincide with the projective tensor product $$B(H_1)\widehat{\otimes}B(H_2)$$ (compare Lemma 4.3.12 and the proof of Proposition 4.3.13. The latter proof can however be fixed, using more von Neumann theory).
It is also slightly strange (if harmless) that the author deliberately refers to quotient maps $$q\colon G\to G/K$$ (with $$G$$ a locally compact group and $$K\subseteq G$$ a closed subgroup) as ‘principal fiber bundles’ (e.g., in Lemma 5.9.3), no matter whether continuous local sections exist or not (as in the case where $$G:={\mathbb T}^{\mathbb N}$$ is a countably infinite power of circle groups and $$K:=\{-1,1\}^{\mathbb N}$$).
Summing up, this book succeeds in providing a point of entry into the universe of harmonic analysis on commutative spaces and leads the reader to the frontiers of current research. Besides a main body of material presented with full (if condensed) proofs, it contains outlooks on more advanced material, which help the reader to get an idea of the area in the large, and make the book a rich source of information also for the specialist. The demands on a real novice are certainly not low – mainly because the ‘crash courses’ are quite condensed. But for those beginners who master them (and for the majority of more advanced readers), the reward will be high.

##### MSC:
 22E30 Analysis on real and complex Lie groups 43A85 Harmonic analysis on homogeneous spaces 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 22D10 Unitary representations of locally compact groups 43A35 Positive definite functions on groups, semigroups, etc. 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis