Representation theory is one of the central themes of modern algebra, and is connected to half a dozen other parts of mathematics. The late George Mackey wrote on several occasions of the important roles played by representation theory in what at first glance might look like rather disparate subjects; see his classic Unitary Representations in Physics, Probability, and Number Theory, for example. Once one has done any reasonable amount of work in the right subdiscipline of any of these, it all starts to look more and more natural: my own favorite example is that of the role played by unitary representation theory in quantum mechanics, i.e. the construction of the Schrödinger representation of the Heisenberg group, and the translation of this construction to local fields (i.e., the localizations of an algebraic number field at its prime valuations, both archimedean and non-archimedean) by André Weil. Central to his famous explication of C. L. Siegel’s analytic theory of quadratic forms, this was part and parcel of Weil’s seminal 1964 Acta Math paper, “Sur certains groupes d’opérateurs unitaires.” One way to see how these two themes, from physics and number theory, are actually all but the same thing is by means of the theorem of Stone and von Neumann, which brings out the crucial part played by central characters: a very beautiful motif.
This material, and, on a larger scale, unitary representation theory, come out of von Neumann’s efforts to make quantum mechanics more mathematically respectable and the all but prophetic work of Weyl on this interphase between physics and analysis. It is not easily accessible to beginners in any of the areas mentioned above: functional analysis is a prerequisite, as are graduate level complex analysis and algebra. To be sure, in algebra proper one does encounter the beautiful and in many ways autonomous subject of representations of finite groups (there is no better book for this than Serre’s Linear Representations of Finite Groups). This material surely translates to the “other kind” of representation theory we are concerned with now, that of infinite groups, i.e. topological groups, but it’s clear as water that, if only by virtue of the presence of non-discrete topologies, the wicket has gotten a great deal stickier.
And this all jives with the fact that the student is often in for a shock when opening for the first times such texts as Alain Robert’s Introduction to the Representation Theory of Compact and Locally Compact Groups (LMS Lect. Notes 80) and the famous treatise Representation Theory and Automorphic Functions by Gel’fand, Graev, and Piatetskii-Shapiro. The representation theory dealt with there is a horse of a different color, really, and can make for some heavy going for the uninitiated. (This was certainly my experience as a graduate student, but the experience was greatly mitigated by another set of LMS Lecture Notes, namely, no. 80: Representation Theory of Lie Groups — and note how vast the subject really is!)
The book under review, by the late Debabrata Basu, is designed to make this transition from the fare of “grad algebra” to the much more analytically and topologically endowed representation theory found in contemporary analytic number theory, quantum physics, as well as certain other living subjects, easier —a lot easier, in fact. As John Klauder describes it in his foreword, “… the author offers a quality introduction and review of important analytical methods and tools in Part I, and in Part II applies them to an important set of Lie groups presenting them in a modern analytical language that also makes close contact with the language of the well known coherent states.”
Thus, the fist 136 pages of the book deal with complex analysis, special functions (Gamma, Beta, zeta, hypergeometric, Bessel), and generalized functions, and, indeed, one recognizes the influence of those Basu refers to as “the Russian masters of functional analysis and representation theory.” Then, on p. 137 (numerologically apt in a book dealing with the indicated theme: viz. the fine structure constant), the rubber hits the road with “Part II: Applications to Group Representation Theory,” which starts off with something of an intensive course on Lie group representations. This discussion, crucial to the whole treatment, of course, includes Schur’s Lemma in two forms, the Peter-Weyl theorem, and then (given the physics angle) “The Clebsch-Gordan Problem and the tensor operators,” followed by a treatment of infinitesimal generators of a representation.
This done, Basu hits examples of great importance in physics: rotations in R3 and SU(2), SU(3), the Lorentz group in three and four dimensions. Spinors figure in the latter two, of course.
And then we get to Chapter 11, the last one: “The Heisenberg-Weyl Group and the Bargmann-Segal Spaces.” As I already indicated above, this material is very close to my heart and I am very happy to see its treatment along these lines, even though the focus is on physics rather than number theory. Given the book’s objective, this is entirely proper, and it is very exciting to encounter a thorough, autonomous discussion of the metaplectic representation (which comes from the projective Weil representation, a.k.a. the Segal-Shale-Weil representation or the oscillator representation: this is truly an ecumenical subject!
Well, enough said. Basu’s book is a very important piece of work — I wish it had been around when I was a graduate student. Two caveats: no exercises, no index. But I do not think this is too much of a problem: go through the book carefully, working things out together with Basu, and you will learn a hell of a lot of important and beautiful mathematics, as well as a chunk of serious modern physics including some of the quarky material for which Gell-Mann and Ne’eman won their Nobel prizes.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.