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Induced representations of locally compact groups. (English) Zbl 1263.22005

Cambridge Tracts in Mathematics 197. Cambridge: Cambridge University Press (ISBN 978-0-521-76226-7/hbk; 978-1-139-04539-1/ebook). xiii, 343 p. (2013).
Induced representations as a means for producing representations of a group \(G\) from those of its subgroups goes back to F. G. Frobenius. It is the core of G. W. Mackey’s program of identifying all irreducible representations of \(G\). A major contribution to this program by J. M. G. Fell deals with the study of the set \(\widehat{G}\) of unitary equivalence classes of irreducible unitary representations as a topological space.
The aim of this nicely written book is to make the theory of induced representations accessible to a wider audience, and a large number of examples facilitates the exposition.
The introductory Chapter 1 presents some basics of harmonic analysis, of the theory of unitary representations (the book deals only with unitary representations) and of \(C^*\)-algebras. The construction of the induced representation is described in Chapter 2, together with standard simplifications in special cases (e.g., for inducing from an open subgroup, or from the normal factor in a semidirect product). The pivotal theorem of the book, the imprimitivity theorem, is proved in Chapter 3.
Chapter 4 deals with the procedure, known as Mackey analysis, for constructing \(\widehat{G}\) for a given locally compact group \(G\). In order for this procedure to work, \(G\) must have a closed normal subgroup \(N\) such that \(\widehat{N}\) is understood, the orbit structure on \(\widehat{N}\), under the action of \(G\), must be well-behaved; and stability subgroups (of \(G/N\)) arising in this action must have a well-understood representation theory. The constructions involved simplify when \(N\) is abelian. To make the concept elementary, the authors start with the case, when \(N\) is abelian of finite index. They finish Chapter 4 with the case of non-abelian \(N\), where \(C^\ast\)-algebraic techniques and the orbit structure of actions on non-Hausdorff spaces are required. Tools for studying the topological structure of \(\widehat{G}\) when the Mackey analysis is successful, are developed in Chapter 5, and a lot of examples are presented.
The first 4–5 chapters make a good textbook for a graduate course in induced representations. The rest of the book illustrates some of (many) ways in which induced representations and knowledge of the topology of \(\widehat{G}\) can be applied. Chapter 6 discusses topological versions of Frobenius properties generalizing the Frobenius reciprocity theorems for finite or compact groups. Chapter 7 explores the asymptotic behavior of the coefficient functions of the irreducible representations of motion groups and methods for constructing projections in \(L^1(G)\).

MSC:

22D10 Unitary representations of locally compact groups
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22D35 Duality theorems for locally compact groups
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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