Representation theory of semisimple groups. An overview based on examples. With a new preface by the author. Paperback ed.

*(English)*Zbl 0993.22001
Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. xix, 773 p. (2001).

The original hardcover edition of this book has been published in 1986 as part of the Princeton Mathematical Series, No. 36. It has been out of print for two years, and the book continues to be in demand. The present edition is the third printing, and the first Princeton Landmarks in Mathematics edition, with a new preface by the author. In this preface, the author specifies two of the fields of application – automorphic forms and analysis of semisimple symmetric spaces – that have undergone remarkable advances since the time of the book’s original publication in 1986, and for which the theory in the book has been indispensable. He also mentions newer fields, such as Kac-Moody algebras and quantum groups, that promise to use more and more of this theory. Finally, it is specified in the preface that the attempts at solving the key problem in Chapter XVI – that of finding all the irreducible unitary representations for all semisimple groups – have led to new approaches and new problems in the field of algebraic groups and geometric group actions. As nothing of the text has been changed in the Landmarks edition, the review Zbl 0604.22001 of the original edition of 1986 gives an adequate description of the contents and structure of the Landmarks edition as well.

Reviewer: Vladimir L. Popov (Moskva)

##### 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. |