##
**Noncommutative harmonic analysis.**
*(English)*
Zbl 0604.43001

Mathematical Surveys and Monographs, 22. Providence, RI: American Mathematical Society (AMS). XVI, 328 p. $ 68.00 (1986).

The fact that harmonic analysis, commutative and noncommutative, is still one of the most active fields of research with strong influences in the applications to physics is reflected by an increasing number of monographs that is going to be published on this important subject. One of the most serious difficulties that the authors are confronted with is the large amount of topics from different areas of mathematics such as multilinear algebra, Lie algebras, modern differential geometry, linear representation theory, functional analysis, linear PDEs, special functions, etc. that have to be dealt with in some detail to prepare the reader. One way to overcome these difficulties in a monograph of reasonable size is to work out a carefully chosen list of specific examples as has been done by A. W. Knapp in his recent monograph [Representation theory of semisimple groups (1986; Zbl 0604.22001)].

The author of the book under review places special emphasis on the rôle played by harmonic analysis in contemporary studies of linear PDEs. His favorite special groups are the nilpotent Lie groups, and especially the Heisenberg Lie group which is two-step nilpotent and therefore the simplest noncommutative noncompact Lie group. This Lie group has the advantage that its representation theory is at once simple and rich in structure and that its various realizations have important applications to quantum mechanics and laser opto-electronics via the theory of coherent and squeezed states. It would have been a valuable help for the reader to point out at least one of the applications. The irreducible unitary linear representations of the Heisenberg nilpotent Lie group are classified in Chapter 1 by the Stone-von Neumann theorem. Unfortunately the author ignores completely the Kirillov coadjoint orbit picture, although this method is extremely useful in the applications. See, for instance, the reviewer’s recent monograph entitled ”Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory” (Pitman Res. Notes Math. 147) Longman, Harlow, Essex, and Wiley & Sons, New York 1986, and the forthcoming subsequent volume.

A study of the Heisenberg nilpotent Lie group introduces in a natural manner other important Lie groups, in particular the symplectic group, acting as a group of automorphisms of the Heisenberg group letting its one-dimensional center pointwise fixed, and the unitary group, arising as a maximal compact subgroup of the symplectic group. Chapter 2 studies the unitary groups U(n), and Chapter 3 contains a number of general results concerning compact Lie groups such as the Peter-Weyl theorem and the Borel-Weil theorem on heighest weight representations. The well-known interplay between the orthogonal group and harmonic analysis on spheres \(S_{n-1}=SO(n)/SO(n-1)\) which provides an elegant approach to the classical theory of spherical harmonics is presented in Chapter 4.

The subsequent Chapter 5 is of central importance since it discusses the inducing procedure for unitary representations and the notion of imprimitivity system. In particular, it establishes the Stone-von Neumann theorem which is a corner-stone of Chapter 1. Chapter 6 deals with harmonic analysis on general nilpotent Lie groups, whereas Chapter 7 is concerned with the solvable group of affine transformations of the real line \({\mathbb{R}}\). The Hankel operator arises as an operator intertwining two irreducible unitary representations of this group. Chapter 8 derives the classification of the irreducible unitary representations of the group SL(2, \({\mathbb{R}})\), Chapter 9 studies SL(2, \({\mathbb{C}})\), whereas Chapter 10 considers the actions of the Lorentz group as groups of conformal transformations on balls and spheres. Chapter 11 studies the oscillator representation of the metaplectic group and Chapter 12 is devoted to a study of spinor groups. The final Chapter 13 forms a very brief introduction to the general theory of noncompact semisimple Lie groups.

The monograph under review forms an elegant PDE oriented presentation of modern noncommutative harmonic analysis written in a condensed and rigorous style. It is based on lecture notes taken from a one semester course taught by the author at Stony Brook. According to its preface, the book is addressed to students with a basic understanding of commutative harmonic analysis and functional analysis. These prerequisites, however, do not suffice. The motivations for the students to master the contents are extremely poor. At least one section dealing with the quantum mechanical harmonic oscillator seems to be necessary to improve the access to the theory and to appreciate its value.

The author of the book under review places special emphasis on the rôle played by harmonic analysis in contemporary studies of linear PDEs. His favorite special groups are the nilpotent Lie groups, and especially the Heisenberg Lie group which is two-step nilpotent and therefore the simplest noncommutative noncompact Lie group. This Lie group has the advantage that its representation theory is at once simple and rich in structure and that its various realizations have important applications to quantum mechanics and laser opto-electronics via the theory of coherent and squeezed states. It would have been a valuable help for the reader to point out at least one of the applications. The irreducible unitary linear representations of the Heisenberg nilpotent Lie group are classified in Chapter 1 by the Stone-von Neumann theorem. Unfortunately the author ignores completely the Kirillov coadjoint orbit picture, although this method is extremely useful in the applications. See, for instance, the reviewer’s recent monograph entitled ”Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory” (Pitman Res. Notes Math. 147) Longman, Harlow, Essex, and Wiley & Sons, New York 1986, and the forthcoming subsequent volume.

A study of the Heisenberg nilpotent Lie group introduces in a natural manner other important Lie groups, in particular the symplectic group, acting as a group of automorphisms of the Heisenberg group letting its one-dimensional center pointwise fixed, and the unitary group, arising as a maximal compact subgroup of the symplectic group. Chapter 2 studies the unitary groups U(n), and Chapter 3 contains a number of general results concerning compact Lie groups such as the Peter-Weyl theorem and the Borel-Weil theorem on heighest weight representations. The well-known interplay between the orthogonal group and harmonic analysis on spheres \(S_{n-1}=SO(n)/SO(n-1)\) which provides an elegant approach to the classical theory of spherical harmonics is presented in Chapter 4.

The subsequent Chapter 5 is of central importance since it discusses the inducing procedure for unitary representations and the notion of imprimitivity system. In particular, it establishes the Stone-von Neumann theorem which is a corner-stone of Chapter 1. Chapter 6 deals with harmonic analysis on general nilpotent Lie groups, whereas Chapter 7 is concerned with the solvable group of affine transformations of the real line \({\mathbb{R}}\). The Hankel operator arises as an operator intertwining two irreducible unitary representations of this group. Chapter 8 derives the classification of the irreducible unitary representations of the group SL(2, \({\mathbb{R}})\), Chapter 9 studies SL(2, \({\mathbb{C}})\), whereas Chapter 10 considers the actions of the Lorentz group as groups of conformal transformations on balls and spheres. Chapter 11 studies the oscillator representation of the metaplectic group and Chapter 12 is devoted to a study of spinor groups. The final Chapter 13 forms a very brief introduction to the general theory of noncompact semisimple Lie groups.

The monograph under review forms an elegant PDE oriented presentation of modern noncommutative harmonic analysis written in a condensed and rigorous style. It is based on lecture notes taken from a one semester course taught by the author at Stony Brook. According to its preface, the book is addressed to students with a basic understanding of commutative harmonic analysis and functional analysis. These prerequisites, however, do not suffice. The motivations for the students to master the contents are extremely poor. At least one section dealing with the quantum mechanical harmonic oscillator seems to be necessary to improve the access to the theory and to appreciate its value.

Reviewer: W.Schempp

### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

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

43A90 | Harmonic analysis and spherical functions |

22E25 | Nilpotent and solvable Lie groups |

22D30 | Induced representations for locally compact groups |

22E30 | Analysis on real and complex Lie groups |