Despite some earlier curricular exposure, I guess I first learned to appreciate the beauty and power of harmonic analysis when I was called to study quantum mechanics more seriously about a decade ago. At that time, my researches in analytic number theory had begun to take a turn in the direction of connections with quantum physics, certainly something unexpected, at least to me. I soon turned to what seemed to me to be the most mathematically accessible presentation of the subject, Prugovecki’s *Quantum Mechanics in Hilbert Space*: I recommend it to anyone who needs to learn QM free of the presuppositions, technical or psychological, the physicists make. My coming to harmonic analysis in this way was more than a little ironic, given my PhD advisor’s expertise in and love for this material. At the time I was her student in the early 1980s, she was working on the early drafts of the following *Harmonic Analysis on Symmetric Spaces *(I still have chapters of her first book in xeroxed form). But I’m afraid that Terras’s focus on harmonic analysis didn’t affect me the way it did some of her subsequent students. I think part of the reason is that my thesis work with her was largely focused on other things, specifically modular forms and Dirichlet series, and harmonic analysis only came in through some relevant representation theory (of modular groups).

That said, it does all testify to my rookie-status at that time: representation theory is of course organically connected to harmonic analysis, and that interplay is in fact a major theme in modern *Zahlentheorie*: nothing less than the Langlands Program brings this to light very spectacularly. So, with enough said about my youthful goofiness, let’s cut to the chase, namely the book under review, two of whose foci are nothing less than number theory and representation theory.

The harmonic analysis properly so-called that arises in the book is described by the authors as coming in two flavors: “Finite commutative harmonic analysis” (dealing with finite abelian groups; often the focus falls on a more general case, i.e. locally compact abelian groups, but we are in the same ballpark), and “Finite harmonic analysis,” which the authors characterize as “representation theory of finite groups: from the basics to \(GL(2,\mathbb{F}_q)\).”

In point of fact, the authors propose the foregoing pair of characterizations in the context of recommendations of autonomous courses for which this book can be used. In addition to the above two suggestions, they add that the book would serve well as a text for a course specifically on finite abelian groups, featuring the discrete and fast Fourier transforms, or as a text for a course on graph theory. The book is split up into four parts which can be tailored to the according needs given these options. The parts are, respectively, “Finite abelian groups and the DFT,” “Finite fields and their characters,” “Graphs and expanders,” and “Harmonic analysis on finite linear groups.”

So it’s clear that the book covers a lot of ground, and should indeed be of great interest to number theorists, fledgling and otherwise. Well, not so fledgling, I suppose:

The final part of the present monograph is devoted to the representation theory of finite groups with emphasis on induced representations and Mackey theory. This includes a complete description of the irreducible representations of the affine groups and Heisenberg groups with coefficients in both … \(\mathbb{F}_q\) and … \(\mathbb{Z}/n\mathbb{Z}\) … In Chapter 13 we develop, with a complete and original treatment, the basic theory of multiplicity-free triplets, their associated spherical functions, and (commutative) Hecke algebras. This is a subject that has not yet received the attention it deserves … The exposition culminates with a complete treatment of the representation theory of \(GL(2,\mathbb{F}_q)\), along the lines developed by Piatetski-Shapiro: our approach, via multiplicity-free triples, constitutes our original contribution to the theory.

So, while the book is written “to be as self-contained as possible,” requiring just linear algebra up to and including the spectral theorem, basic group and ring theory, and “elementary number theory,” the reader is exposed to a lot of serious mathematics, some even at or near the frontier. For one thing, the authors present “Dirichlet’s theorem on primes in arithmetic progressions, which is based on the character theory of finite abelian groups” (see Chapter 3 in Part I). For another, they present “Tao’s uncertainty principle for (finite) cyclic groups” (Chapter 2).” So, indeed, get on your flight suits, and strap in.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.