##
**An introduction to harmonic analysis on semisimple Lie groups.**
*(English)*
Zbl 0753.22003

Cambridge Studies in Advanced Mathematics. 16. Cambridge: Cambridge University Press. x, 316 p. (1989).

This book is designed as an introduction to the subject of harmonic analysis on semisimple Lie groups, nominally at the level of advanced undergraduate – beginning graduate. This however is to be viewed as a description of mathematical prerequisites for reading the book – the author of these words is strongly convinced that the book can be an extremely valuable source of information for specialists in other branches of mathematics willing to get an orientation in this fascinating but vast and highly specialized subject. It might be said that the appearance of the book marks a certain level of maturity achieved by the subject itself. Here is an authoritative account of the main, classical themes of the theory, placed in the historical perspective of their development and containing most of its basic notions. The main techniques of proofs are outlined, most often presented on a carefully chosen particular case rather than in its full generality, in accordance with the introductory character of the book. And all this is contained in a handy, slim volume of slightly more than three hundred pages!

The book is divided into eight chapters: 1) Introduction, 2) Compact groups: the work of Weyl, 3) Unitary representations of locally compact groups, 4) Parabolic induction, principal series representations and their characters, 5) Representations of Lie algebra, 6) The Plancherel formula: character form, 7) Invariant eigendistributions, 8) Harmonic analysis of the Schwartz space, which are supplemented by three appendices describing prerequisites from functional analysis, topological groups and Lie groups and Lie algebras resp. The first three chapters contain introductory material and begin with a brief description of the roots of the subject in the classical Fourier analysis, algebraic theory of invariants, arithmetic and also quantum mechanics. Subsequently the author passes to a brief presentation of elements of the work of Hermann Weyl on representations of compact semisimple Lie groups including the Peter-Weyl theory, description of characters of irreducible representations in both infinitesimal and global formulation and the unitarian trick. A discussion of the Plancherel formula in both the classical Fourier analysis and the analysis on compact Lie groups enhances the readers understanding, why establishing of this formula is regarded as the central issue of \(L^ 2\) harmonic analysis.

This part concludes with brief presentations of essentials of Mackey’s general theory of induced representations of locally compact topological groups, including a description of his construction of irreducible representations of semidirect products. The following chapter introduces the main subject of the book, approached here in the historical perspective shaped mainly by the pathbreaking papers of Harish-Chandra. A brief discussion of the early attempts at constructing the representation theory of complex classical Lie groups in the work of Gelfand and Naimark serves to motivate the presentation of the guiding principles of Harish- Chandra’s strategy for developing the \(L^ 2\) harmonic analysis. These were: firstly, the idea of constructing irreducible representations of \(\mathcal G\) needed for obtaining Plancherel formula by inducing suitable representations of its parabolic subgroups associated to various classes of Cartan subgroups (the so called parabolic induction) and, secondly, the key role played in the construction by the so called discrete series representations of certain semisimple subgroups of \(\mathcal G\) (those occurring in the Levy decomposition of parabolic subgroups). Thus from Harish-Chandra’s perspective the development of the theory rests on achieving two fundamental goals: 1) Constructing the discrete series for \(\mathcal G\) when it has a compact Cartan subgroup. 2) Proving the completeness of the set of irreducible representations obtained by the method of restricted parabolic induction. The use of the parabolic induction is illuminated by a fairly detailed description of the principal series representations and the calculation of their characters in the case of \(SL(n,C)\).

Chapter 5 describes representations of semisimple Lie algebras and their relationship to representations of semisimple Lie groups. Thus properties of the Lie algebra representation on the space of analytic vectors for a given representation of the corresponding group are investigated. This leads to an algebraic description of the fundamental objects of study in representation theory by introducing an appropriate category of Lie algebra modules – the Harish-Chandra modules and to a discussion of their main properties. Even if some of the deeper results of the theory are not proved, like the subquotient theorem of Harish-Chandra, their meaning is always clearly explained and the included proofs provide the reader with substantial insight into the technique employed in the theory. The chapter ends by a detailed description of Harish-Chandra modules for \(SL(2,R)\), and a construction of the supplementary series for \(SL(2,C)\). The following chapter explains ideas of Harish-Chandra’s approach to the character form of the Plancherel formula – first for complex groups (here the case of \(SL(n,C)\) is worked out in detail) and the extension for real groups is discussed at the case of \(SL(2,R)\). The reader learns here about the techniques of transferring orbital integrals from the group to its Lie algebra, the limit formula, radial parts of invariant differential operators and is shown finally a proof of the Plancherel formula for \(SL(2,R)\). A short Chapter 7 brings a proof of Harish-Chandra’s theorem on regularity of invariant eigenfunctions, given in the case of \(SL(2,R)\).

The final chapter 8 is the longest and the deepest one in the book and it leads into the more profound aspects of the \(L^ 2\) harmonic analysis, accessible through the study of the Schwartz space on \(\mathcal G\). Although everything in this chapter is done for the case \({\mathcal G}=SL(2,R)\), the notation is general and as previously the techniques are those used in the general context of an arbitrary semisimple Lie group with finite centre. The main themes of the chapter are the asymptotic properties of matrix elements, determination of the Plancherel measure using the asymptotic properties of eigenfunction expansion, wave packets and properties of the Harish-Chandra transform.

The book is divided into eight chapters: 1) Introduction, 2) Compact groups: the work of Weyl, 3) Unitary representations of locally compact groups, 4) Parabolic induction, principal series representations and their characters, 5) Representations of Lie algebra, 6) The Plancherel formula: character form, 7) Invariant eigendistributions, 8) Harmonic analysis of the Schwartz space, which are supplemented by three appendices describing prerequisites from functional analysis, topological groups and Lie groups and Lie algebras resp. The first three chapters contain introductory material and begin with a brief description of the roots of the subject in the classical Fourier analysis, algebraic theory of invariants, arithmetic and also quantum mechanics. Subsequently the author passes to a brief presentation of elements of the work of Hermann Weyl on representations of compact semisimple Lie groups including the Peter-Weyl theory, description of characters of irreducible representations in both infinitesimal and global formulation and the unitarian trick. A discussion of the Plancherel formula in both the classical Fourier analysis and the analysis on compact Lie groups enhances the readers understanding, why establishing of this formula is regarded as the central issue of \(L^ 2\) harmonic analysis.

This part concludes with brief presentations of essentials of Mackey’s general theory of induced representations of locally compact topological groups, including a description of his construction of irreducible representations of semidirect products. The following chapter introduces the main subject of the book, approached here in the historical perspective shaped mainly by the pathbreaking papers of Harish-Chandra. A brief discussion of the early attempts at constructing the representation theory of complex classical Lie groups in the work of Gelfand and Naimark serves to motivate the presentation of the guiding principles of Harish- Chandra’s strategy for developing the \(L^ 2\) harmonic analysis. These were: firstly, the idea of constructing irreducible representations of \(\mathcal G\) needed for obtaining Plancherel formula by inducing suitable representations of its parabolic subgroups associated to various classes of Cartan subgroups (the so called parabolic induction) and, secondly, the key role played in the construction by the so called discrete series representations of certain semisimple subgroups of \(\mathcal G\) (those occurring in the Levy decomposition of parabolic subgroups). Thus from Harish-Chandra’s perspective the development of the theory rests on achieving two fundamental goals: 1) Constructing the discrete series for \(\mathcal G\) when it has a compact Cartan subgroup. 2) Proving the completeness of the set of irreducible representations obtained by the method of restricted parabolic induction. The use of the parabolic induction is illuminated by a fairly detailed description of the principal series representations and the calculation of their characters in the case of \(SL(n,C)\).

Chapter 5 describes representations of semisimple Lie algebras and their relationship to representations of semisimple Lie groups. Thus properties of the Lie algebra representation on the space of analytic vectors for a given representation of the corresponding group are investigated. This leads to an algebraic description of the fundamental objects of study in representation theory by introducing an appropriate category of Lie algebra modules – the Harish-Chandra modules and to a discussion of their main properties. Even if some of the deeper results of the theory are not proved, like the subquotient theorem of Harish-Chandra, their meaning is always clearly explained and the included proofs provide the reader with substantial insight into the technique employed in the theory. The chapter ends by a detailed description of Harish-Chandra modules for \(SL(2,R)\), and a construction of the supplementary series for \(SL(2,C)\). The following chapter explains ideas of Harish-Chandra’s approach to the character form of the Plancherel formula – first for complex groups (here the case of \(SL(n,C)\) is worked out in detail) and the extension for real groups is discussed at the case of \(SL(2,R)\). The reader learns here about the techniques of transferring orbital integrals from the group to its Lie algebra, the limit formula, radial parts of invariant differential operators and is shown finally a proof of the Plancherel formula for \(SL(2,R)\). A short Chapter 7 brings a proof of Harish-Chandra’s theorem on regularity of invariant eigenfunctions, given in the case of \(SL(2,R)\).

The final chapter 8 is the longest and the deepest one in the book and it leads into the more profound aspects of the \(L^ 2\) harmonic analysis, accessible through the study of the Schwartz space on \(\mathcal G\). Although everything in this chapter is done for the case \({\mathcal G}=SL(2,R)\), the notation is general and as previously the techniques are those used in the general context of an arbitrary semisimple Lie group with finite centre. The main themes of the chapter are the asymptotic properties of matrix elements, determination of the Plancherel measure using the asymptotic properties of eigenfunction expansion, wave packets and properties of the Harish-Chandra transform.

Reviewer: A.Strasburger (Warszawa)

### MSC:

22E46 | Semisimple Lie groups and their representations |

22E30 | Analysis on real and complex Lie groups |

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

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

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

43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |

43A80 | Analysis on other specific Lie groups |