*(English)*Zbl 0725.20028

This excellently written book is an advanced textbook on the theory of Coxeter groups. It pursues two objects. Firstly, it is an introduction to the book by *N. Bourbaki* on Lie groups and algebras (chapters 4-6, 1968; Zbl 0186.330). Secondly, it is an updating of the coverage. Correspondingly, the book is divided into two parts.

The first part consists of 4 chapters: finite reflection groups, classification of finite reflection groups, polynomial invariants of finite reflection groups, affine reflection groups.

The second part is inspired especially by the seminal work by *D. Kazhdan* and *G. Lusztig* [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] on representations of Hecke algebras associated with Coxeter groups. This part consists of 4 chapters: Coxeter groups (here there is the Bruhat ordering), special cases, Hecke algebras and Kazhdan-Lusztig polynomials, complements (this chapter sketches a number of interesting complementary topics as well as connections with Lie theory). The book has an extensive bibliography on Coxeter groups and their applications.

##### MSC:

20F55 | Reflection groups; Coxeter groups |

20G05 | Representation theory of linear algebraic groups |

20-02 | Research monographs (group theory) |

51F15 | Reflection groups, reflection geometries |

20H15 | Other geometric groups, including crystallographic groups |