Linear Analysis and Representation Theory

When I started my thesis, in the early 1980s, it soon became clear that I needed to learn a good deal of unitary representation theory at the speed of light. This was a relatively early exposure to one of the core realities of doing mathematical research, namely that the questions and problems one is dealing with dictate what one learns, and a linear approach is generally inimical to decent progress. Thus, one hits half a dozen sources at once, jumping all over the place, sometimes skipping forward, often going back to more basic material to get a better understanding and fill in the gaps. To learn representation theory, in my case of SL(2), PSL(2), and various of their subgroups of number theoretic significance, I recall checking several books out of the UCSD library and haunting bookstores in La Jolla and its surroundings until, Deo gratias, I came across a gem titled Representation Theory of Lie Groups, an LMS Lecture Notes Series entry (no. 34), edited by Michael Atiyah et al. Perfect! I still cherish my beat-up copy, with my thirty-year-old marginalia all over the pages. A happy ending: together with three or four other books, LMS 34 did what it had to do to cover the representation theoretic parts of my dissertation.

How much easier it would have been if I had owned a copy of this book! It wasn’t as though it didn’t exist: Steven Gaal published his first edition of Linear Analysis and Representation Theory in 1973. It first appeared as a Springer-Verlag publication, the present Dover version being an unabridged republication of the original. As a Springer “Yellow Peril” book, it was doubtless quite pricey and therefore out of reach; also, I guess in my naiveté I was looking at much narrower titles. Happily the present Dover version of this fantastic book lists for less that thirty dollars: a steal.

It’s a beefy book: twelve pages shy of 700 pages, but it covers a lot of great stuff. After two chapters on Banach algebras and operator algebras, the book’s central themes are developed in order: spectral theory, elementary representation theory in Hilbert space, topological groups, induced representations, and finally a pair of chapters devoted to square-integrable representations, spherical functions, trace functions, and Lie theory. Truly wonderful material. The only book I can think of that qualifies as a competitor to Gaal’s is the classic by Gel’fand, Graev, and Piatetskii-Shapiro, Representation Theory and Automorphic Forms (this clearly betrays my number-theoretic bias). But it’s worth noting that according to Wikipedia, Gaal was brought to IAS by Atle Selberg and that Robert Langlands cites Gaal as one of his influences vis à vis his work on zeta functions and Eisenstein series. Fledgling arithmeticians and modular formers, take note!

Given the fact that the text is now almost 40 years old, there is little or nothing avant garde in the book. If you’re in a hurry to get to the frontier of current work on representation theory, Gaal is not the way to go. But it is a phenomenal source for the indispensable foundation, going very deep in an extremely elegant manner. Gaal’s opening phrases are still quite pithy:

In an age when more and more items are made to be quickly disposable or soon become obsolete due to either progress or other man caused reasons it seems almost anachronistic to write a book in the classical sense. A mathematics book becomes an indispensable companion, if it is worthy of such a relation, not by being rapidly read from cover to cover but by frequent browsing, consultation, and other occasional use.

I think this hits the mark.

One final caveat: Linear Analysis and Representation Theory is not a textbook: there are no lists of problems, and there are no motivational examples. On the other hand, Gaal appends to each chapter a section titled “Remarks,” and these (actually pretty compact) sections serve to give the reader both a broader and historically based perspective on the material and a road map to other important sources.

Linear Analysis and Representation Theory is high scholarship devoted to a beautiful and centrally important subject. This is one of those books that “everyone” should own (modulo having some need of doing representation theory beyond that of finite groups).

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