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Harmonic Analysis on Symmetric Spaces — Higher Rank Spaces, Positive Definite Matrix Space and Generalizations

Audrey Terras
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
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Some two-and-a-half years ago I had the pleasure of reviewing, in this column, the first volume in this series of books by Audrey Terras, whose PhD student I was in the early 1980s. The book under review obviously takes off where its predecessor left off, and its subtitle, indicated above, gives a good indication of its focus; the present book is in fact the second edition of the corresponding work Audrey launched in 1987. Here is a phrase from her Preface to the first edition, distinctive indeed, and indicative of the idiosyncratic style of Audrey’s writing: “Well, finally here it is — the long-promised revenge of the Higher Rank Symmetric Spaces and Their Fundamental Domains. When I began work on it in 1977, I would probably have stopped immediately if someone had told me that 10 years would pass before I would declare it ‘finished.’ Yes, I am declaring it finished — though certainly not perfected. There is a large amount of work going on at the moment as the piles of preprints reach the ceiling. Nevertheless it is summer and the ocean calls …” Ah, yes, the perks of being at UCSD in La Jolla.

The present volume is hot off the presses and accordingly Audrey’s Preface to the Second Edition dates to August of 2015. Says Audrey: “It is marvelous and a bit scary to return to this garden … Sadly many people such as Serge Lang, Hans Maass, and Atle Selberg are now gone. But there are many new flowers …” and she goes on to mention a number of recent sources and then states that “[t]hanks to younger people, there are [now] even computations of automorphic forms for \(\mathrm{GL}(n,\mathbf{Z})\), when \(n=3\) and \(4\).” Additionally, “[t]here is much new work on random matrices. And of course there are many adèlic books and papers …” Indeed, the book’s list of references weighs in at a daunting 677 entries, giving evidence of precisely what Audrey is talking about: it’s a beehive out there.

A few comments, then, about what has been added to this Second Edition: “… a few new sections, including one on Donald St. P. Richards’ central limit theorem for \(\mathrm{O}(n)\) -invariant random variables on the symmetric space of \(\mathrm{GL}(n,\mathbf{R})\), another on random matrix theory, and some discussions of mostly non-adèlic advances in the theory of automorphic forms on arithmetic groups since 1987.” This said, the book under review is split up into two long chapters, the first dealing with the space of positive \(n\times n\) matrices, culminating in a discussion of Maass forms for \(\mathrm{GL}(n,\mathbf{Z})\) and attendant harmonic analysis, and the second focused on the case of general non-compact (!) symmetric spaces, appropriately culminating in a discussion of fundamental domains, automorphic forms, and trace formulas: the life’s blood of this whole business, of course, at least as far as number theory is concerned.

Regarding prerequisites for the book, Audrey suggests that the reader should be sufficiently well-prepared with a decent grounding in differential geometry and “a little knowledge of the beginning of algebraic number theory.” So, if I may elaborate (and improvise) a bit, I’d suggest the contents of Loring Tu’s excellent and eminently readable Introduction to Manifolds and J. S. Milne’s lectures on algebraic number theory available here. Of course, given the presence of Hecke and Lang in the game is Audrey is playing, there are also the following books to consider, but let’s say they are a bit more, well, again, idiosyncratic: Lang’s Algebraic Number Theory and Hecke’s (in my opinion altogether unparalleled) Lectures on the Theory of Algebraic Numbers.

It is very nice to have available, now, the second, updated version of the entire set, volumes one and two, of this important scholarship on the theory of symmetric spaces and the harmonic analysis that is done on them. Audrey Terras has done the mathematical community (and not just number theorists and modular formers) a great service: these books are a major contribution on several fronts, including the pedagogical one. They are of course also excellent references for various mathematical themes that are otherwise scattered all through the recent literature.

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