##
**Representation theory of semisimple groups. An overview based on examples.**
*(English)*
Zbl 0604.22001

Princeton Mathematical Series, 36. Princeton, New Jersey: Princeton University Press. XVII, 773 p. $ 75.00 (1986).

This book is the manual on the representation theory of semisimple Lie groups, including the main results and techniques of the theory. It is organized in such a way that corresponds better to a reader’s learning process, where ideas, problems and examples are more important than details of complicated proofs.

In the first part of the book (Chapters 1-5) the author gives an introduction into the representation theory of semisimple Lie groups. Chapter 1 contains basic definitions and simple results on the structure and representations of Lie groups. In Chapter 2 the reader meets the finite dimensional and unitary irreducible representations of the groups SL(2, \({\mathbb{C}})\) and SL(2, \({\mathbb{R}})\). Transition from the representation of a Lie group G in the space E to the representation of the universal enveloping algebra U(\({\mathfrak g})\) of the Lie algebra \({\mathfrak g}\), \({\mathfrak g}=Lie(G)\) in the subspace of \(C^{\infty}\)-vectors of this space E and vice versa is the subject of Chapter 3. The representation theory of compact Lie groups is given in Chapter 4 in full generality, its results are used throughout the text.

The main body of the book begins with Chapter 5, where the principal facts on the structure of semisimple Lie groups are collected. Chapter 6 contains the description of the holomorphic discrete series of representations of real semisimple Lie groups. In Chapter 7 basic techniques of representation theory: induced representation, intertwining operators, spherical functions appear. Connection between spherical functions and the size of matrix coefficients is established, and in the end of this Chapter the hero of the classification problem - the Langlands quotient - is defined.

Chapter 8 is the first kernel of the book. The action of the center Z(\({\mathfrak g})\) of the enveloping algebra U(\({\mathfrak g})\) on the subspace of K-finite vectors of a G-module is studied here (in context of admissible representations). This leads to the investigation of the differential equations, satisfied by matrix coefficients of the representation. The results on the asymptotic size and leading exponents of matrix coefficients, thus obtained, permit to prove the subrepresentation theorem in the form of Casselman, to study the analytic continuation of intertwining operators, to control the K-finite Z(\({\mathfrak g})\)-finite functions on the group G. They also permit to show that the leading exponents of irreducible admissible representations control the size of the matrix coefficients and to deduce the characterization of discrete series and tempered irreducible representations in terms of leading exponents or in terms of UIR, induced from the unitary irreducible representations (UIR) \(\omega \otimes e^{\nu}\) of the parabolic subgroup \(S=MAN\), with discrete series representation \(\omega\) and imaginary parameter \(\nu\). Finally, the main result of the Chapter - the Langlands classification theorem - is proved.

Description of discrete series representations - the main ingredient of the Langlands classification - and the discussion on the proof of the Plancherel formula is the content of Chapters 9-13. Remark that the construction of discrete series in Chapter 9 is extracted from the work of M. Flensted-Jensen on the discrete series on semisimple symmetric spaces. Exposition of properties of global characters of irreducible admissible representations is given in Chapter 10. Behavior of the character on the set of regular points of the group G and that under tensoring on the finite-dimensional G-module is studied here. In Chapter 12, after discussion on the exhaustion of discrete series, the author studies the limits of discrete series.

Chapter 14 is the second kernel chapter of the book. The classification of irreducible tempered representations (due to Knapp and Zuckerman) is given here. The main tool are the results on the reducibility of the representations, induced from the parabolic subgroup MAN with a discrete series on M and unitary character on A. The intertwining operators play a pivotal role here, so, the author studies the structure of the ring of intertwining operators by using the asymptotics of Eisenstein integrals, connected with the matrix coefficients. Finally, basic characters are introduced and the complete reduction of the induced representations, described above, is given in terms of irreducible basic characters. This leads to the classification of tempered irreducible representations which concludes the solving of the classification problem of admissible irreducible representations.

The theory of minimal K-types is discussed in Chapter 15, where the proof of Langlands’ disjointness theorem is given.

Chapter 16 is concerned with the classification problem of unitary irreducible representations. Some results on the complementary series are given here. The connection of the unitarity of the representation with the positivity of intertwining operators is studied and the chapter is concluded with the reduction of the classification problem to the case of real infinitesimal characters.

Problems are given at the end of each chapter which help the student to learn the material. All the absent proofs and the most general formulations of theorems may be found with the help of the historical and bibliographical notes at the end of the book.

This book is an excellent manual on the representation theory of semisimple Lie groups, this difficult but beautiful and important subject.

In the first part of the book (Chapters 1-5) the author gives an introduction into the representation theory of semisimple Lie groups. Chapter 1 contains basic definitions and simple results on the structure and representations of Lie groups. In Chapter 2 the reader meets the finite dimensional and unitary irreducible representations of the groups SL(2, \({\mathbb{C}})\) and SL(2, \({\mathbb{R}})\). Transition from the representation of a Lie group G in the space E to the representation of the universal enveloping algebra U(\({\mathfrak g})\) of the Lie algebra \({\mathfrak g}\), \({\mathfrak g}=Lie(G)\) in the subspace of \(C^{\infty}\)-vectors of this space E and vice versa is the subject of Chapter 3. The representation theory of compact Lie groups is given in Chapter 4 in full generality, its results are used throughout the text.

The main body of the book begins with Chapter 5, where the principal facts on the structure of semisimple Lie groups are collected. Chapter 6 contains the description of the holomorphic discrete series of representations of real semisimple Lie groups. In Chapter 7 basic techniques of representation theory: induced representation, intertwining operators, spherical functions appear. Connection between spherical functions and the size of matrix coefficients is established, and in the end of this Chapter the hero of the classification problem - the Langlands quotient - is defined.

Chapter 8 is the first kernel of the book. The action of the center Z(\({\mathfrak g})\) of the enveloping algebra U(\({\mathfrak g})\) on the subspace of K-finite vectors of a G-module is studied here (in context of admissible representations). This leads to the investigation of the differential equations, satisfied by matrix coefficients of the representation. The results on the asymptotic size and leading exponents of matrix coefficients, thus obtained, permit to prove the subrepresentation theorem in the form of Casselman, to study the analytic continuation of intertwining operators, to control the K-finite Z(\({\mathfrak g})\)-finite functions on the group G. They also permit to show that the leading exponents of irreducible admissible representations control the size of the matrix coefficients and to deduce the characterization of discrete series and tempered irreducible representations in terms of leading exponents or in terms of UIR, induced from the unitary irreducible representations (UIR) \(\omega \otimes e^{\nu}\) of the parabolic subgroup \(S=MAN\), with discrete series representation \(\omega\) and imaginary parameter \(\nu\). Finally, the main result of the Chapter - the Langlands classification theorem - is proved.

Description of discrete series representations - the main ingredient of the Langlands classification - and the discussion on the proof of the Plancherel formula is the content of Chapters 9-13. Remark that the construction of discrete series in Chapter 9 is extracted from the work of M. Flensted-Jensen on the discrete series on semisimple symmetric spaces. Exposition of properties of global characters of irreducible admissible representations is given in Chapter 10. Behavior of the character on the set of regular points of the group G and that under tensoring on the finite-dimensional G-module is studied here. In Chapter 12, after discussion on the exhaustion of discrete series, the author studies the limits of discrete series.

Chapter 14 is the second kernel chapter of the book. The classification of irreducible tempered representations (due to Knapp and Zuckerman) is given here. The main tool are the results on the reducibility of the representations, induced from the parabolic subgroup MAN with a discrete series on M and unitary character on A. The intertwining operators play a pivotal role here, so, the author studies the structure of the ring of intertwining operators by using the asymptotics of Eisenstein integrals, connected with the matrix coefficients. Finally, basic characters are introduced and the complete reduction of the induced representations, described above, is given in terms of irreducible basic characters. This leads to the classification of tempered irreducible representations which concludes the solving of the classification problem of admissible irreducible representations.

The theory of minimal K-types is discussed in Chapter 15, where the proof of Langlands’ disjointness theorem is given.

Chapter 16 is concerned with the classification problem of unitary irreducible representations. Some results on the complementary series are given here. The connection of the unitarity of the representation with the positivity of intertwining operators is studied and the chapter is concluded with the reduction of the classification problem to the case of real infinitesimal characters.

Problems are given at the end of each chapter which help the student to learn the material. All the absent proofs and the most general formulations of theorems may be found with the help of the historical and bibliographical notes at the end of the book.

This book is an excellent manual on the representation theory of semisimple Lie groups, this difficult but beautiful and important subject.

Reviewer: S.Prishchepionok

### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22E46 | Semisimple Lie groups and their representations |

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

22E30 | Analysis on real and complex Lie groups |

43A90 | Harmonic analysis and spherical functions |

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