Gratia von Neumann, Hermann Weyl, and a few physicists, it’s probably true that the right way for mathematicians to develop quantum mechanics is in functional analytic terms, with the unitary operators featured there requiring a stiff dose of representation theory, and right off the bat, we’re face-to-face with locally compact groups. From another point of view, the theory of the metaplectic group, developed by André Weil in the 1960s, is built around the unitary representation theory of the (locally compact) symplectic group, and this work, which is to a large degree an exploitation of the Stone-Von Neumann theorem, is central to the modern theory of metaplectic forms. Interestingly, a theme common to quantum mechanics and the inner life of the metaplectic group is the centrality of the Heisenberg group (in different flavors), and certain of its induced representations. This, indeed, is a dramatic illustration of a very striking and somewhat mysterious phenomenon, namely, that methodologically as well as structurally quantum physics and number theory are deeply intertwined, if the reader will excuse a cheap pun.
The point is that induced representations of locally compact groups are of great importance and deep significance both to mathematicians and to (certain) physicists, and therefore a text such as the one under review is very desirable indeed. There are a number of sources available that deal with aspects of this material, including of course Weil’s original L’Intégration dans les Groupes Topologiques et ses Applications and Alain Robert’s Introduction to the Representation Theory of Compact and Locally Compact Groups. These books are, in a sense, non-negotiable as far as background in this subject goes: they set the stage for, e.g., Armand Borel’s terse Representations de Groupes Localement Compacts, which is by no means a leisurely introduction to the subject. Indeed, pedagogically speaking, it is Robert’s book which is the front-runner when it comes to preparing the reader for what Kaniuth and Taylor offer in the book under review. But it should be noted that Robert’s discussion of induced representations per se only occupies around seven pages, although the process of induction is prominently featured elsewhere in his book.
Of course, in Kaniuth-Taylor, induced representations take central stage, and this is reasonable in light of the algebraic subtext of the authors’ discussion, exemplified by Mackey’s “machine.” This phrase, or, equivalently, “Mackey analysis” has to do with describing the dual group to a locally compact group from data about the dual group to a closed normal subgroup of the given group. Obviously induction is at the heart of all this.
Mackey analysis is also literally the central chapter of this book. It is preceded by three chapters, respectively on “basics,” a ramified discussion of induced representations, and Mackey’s famous imprimitivity theorem to the effect that a system of imprimitivity for a locally compact group in the presence of a closed subgroup comes from some unitary representation of the subgroup. Then, in the wake of Mackey analysis, topological questions are addressed, particularly the matter of topologies on indicated dual spaces, and the book ends with “further applications.” Says the back cover in this connection: “An extensive introduction to Mackey analysis is applied to compute dual spaces for a wide variety of examples.” Indeed.
Obviously this text is meant for, at least, advanced graduate students headed in the direction of representation theory of locally compact groups, and it is pitched at a professional level, as is consonant with the goals of the Cambridge Tracts in Mathematics. This is good and serious mathematics, presented well, and constitutes a valuable contribution to the discipline.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
2. Induced representations
3. The imprimitivity theorem
4. Mackey analysis
5. Topologies on dual spaces
6. Topological Frobenius properties
7. Further applications