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Non-Abelian harmonic analysis. Applications of \(SL(2,{\mathbb{R}})\). (English) Zbl 0768.43001
Universitext. New York etc.: Springer-Verlag. xv, 257 p. (1992).
Although this book appears in the Springer “Universitext” series, this introduction to representation theory and some of its applications is more likely to find favour amongst postgraduates than undergraduates, since it requires quite a breadth of sophistication.
The book contains five chapters. The first is a fast introduction to the necessary background: from algebra, some ideas of the theory of Lie groups and Lie algebras, and from analysis, notions of Fourier analysis and spectral theory.
The second chapter involves a description of (some of) the indecomposable, rather than irreducible, representations of the Lie algebra \({\mathfrak sl}(2)\). Some of these come as a surprise: they are not submodules of the principal series, and so may well be unfamiliar even to those who have been exposed to \(SL(2,\mathbb{R})\) or more general semisimple groups. Another feature of Chapter 2 is that “formal eigenfunctions” are used: if \(H_ 1,X_ 1,Y_ 1\) and \(H_ 2,X_ 2,Y_ 2\) are different standard bases of \({\mathfrak sl}(2)\), then a module which is an algebraic direct sum of eigenvectors for \(H_ 1\) will not usually contain eigenvectors for \(H_ 2\); formal infinite sums of the eigenvectors for \(H_ 1\) must be allowed.
The third chapter builds on the results of Chapter 2 to obtain a classification of the irreducible unitary representations of \(SL(2,\mathbb{R})\) and its universal covering group, following Bargmann’s original approach, and introduces the oscillator (also well known as Weil) representation. The key fact, for the applications in the following chapter, is that \(\widetilde{SL}(2,\mathbb{R})\) has a unitary representation \(\pi\) on \(L^ 2(\mathbb{R}^ n)\) (or more generally \(L^ 2(\mathbb{R}^{p+q}))\), which commutes with the action of the orthogonal group \(O(n)\) (or the group \(O(p,q)\)), for which the Fourier transform appears as \(\pi(k)\) for a particular element \(k\) of \(\widetilde{SL}(2,\mathbb{R})\).
Chapter 4 shows how representation theory can be used to shed new light on various results of classical analysis, such as Bochner’s periodicity theorem for the Fourier transform, or the formula for the fundamental solutions for the Laplace and wave operators, and also establishes several results of Harish-Chandra, including a version of his celebrated theorem on the local integrability of characters, by analysing \(SO(2,1)\)- invariant eigendistributions (for the invariant wave operator) on \(\mathbb{R}^ 3\).
Finally, in Chapter 5, matrix coefficients of unitary representations and their decay are discussed. It is shown that \(SL(2,\mathbb{R})\) has matrix coefficients which decay very slowly at infinity, while for \(SL(n,\mathbb{R})\), matrix coefficients of non-trivial representations decay in a well controlled manner. This is a quantitive version of Kazhdan’s “property T”.
It is a remarkable feat to fit so much into so little space (250 pages). A good deal of detail is omitted, but the references provided are sufficient to allow the gaps to be filled. The introduction suggests this book might be read after Lang’s \(SL(2,\mathbb{R})\) and before Knapp’s or Wallach’s book in a program to learn semisimple analysis (this is perhaps confirmation that this text is not for the average undergraduate). However, as the description of the contents suggests, the authors of this book are more prone to the digression than Lang, Knapp or Wallach. Some readers will see this as a fault, and others will find it to be a strength, depending on their views on mathematics. This reviewer would encourage the authors to continue to cultivate their own gardens.

43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
22-02 Research exposition (monographs, survey articles) pertaining to topological groups