Unitary representations and harmonic analysis. An introduction. 2nd ed.

*(English)*Zbl 0697.22001
North-Holland Mathematical Library, 44. Amsterdam etc.: North-Holland; Tokyo: Kodansha Ltd. xv, 452 p. $ 112.75; Dfl. 220.00 (1990).

This is a new edition of the excellent book of Mitsuo Sugiura. The first edition has been published in 1975 (cf. Zbl 0344.22001). It is well known by many mathematicians, and it is especially useful for young mathematicians who find in this book a nice introduction in representation theory.

The books begins with a chapter about Fourier series. In the second chapter one studies the groups SU(2) and SO(3) and their representations. The last section is a short presentation of representation theory for compact groups. The Fourier analysis on \({\mathbb{R}}^ n\) is the topic of chapter III. Then in chapter IV one studies the representations of the Euclidean motion group, together with the induction of representations. The last chapter, whose volume is about half of the book, is devoted to the group SL(2,\({\mathbb{R}})\) and its representations.

The second edition differs from the first one at some points. In particular one describes in chapter V of the new edition the limits of the discrete series for SU(1,1), and a section has been added about representations of SL(2,\({\mathbb{R}})\). In fact in the first edition one considered only the realizations of the representations corresponding to the action of SU(1,1) on the unit disc. In the new edition one describes also the realizations of the representations of SL(2,\({\mathbb{R}})\) related to the action on the upper half-plane.

The bibliography has been brought up to date and contains more than seven hundred titles. It is an excellent source of references on harmonic analysis and will be valuable for all researchers in this field.

The books begins with a chapter about Fourier series. In the second chapter one studies the groups SU(2) and SO(3) and their representations. The last section is a short presentation of representation theory for compact groups. The Fourier analysis on \({\mathbb{R}}^ n\) is the topic of chapter III. Then in chapter IV one studies the representations of the Euclidean motion group, together with the induction of representations. The last chapter, whose volume is about half of the book, is devoted to the group SL(2,\({\mathbb{R}})\) and its representations.

The second edition differs from the first one at some points. In particular one describes in chapter V of the new edition the limits of the discrete series for SU(1,1), and a section has been added about representations of SL(2,\({\mathbb{R}})\). In fact in the first edition one considered only the realizations of the representations corresponding to the action of SU(1,1) on the unit disc. In the new edition one describes also the realizations of the representations of SL(2,\({\mathbb{R}})\) related to the action on the upper half-plane.

The bibliography has been brought up to date and contains more than seven hundred titles. It is an excellent source of references on harmonic analysis and will be valuable for all researchers in this field.

Reviewer: J.Faraut

##### MSC:

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

43-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |