Since its appearance in 1968, Bourbaki [1] (treating Coxeter groups, Tits systems, reflection groups, and root systems) has become indispensable to all students of semisimple Lie theory. An enormous amount of information is packed into relatively few pages, including detailed descriptions of the individual root systems and a vast assortment of challenging ‘exercises’. My own dog-eared copy (purchased at Dillon's in London in the spring of 1969 for 90 shillings) is always at hand. The present book attempts to be both an introduction to Bourbaki and an updating of the coverage, by inclusion of such topics as Bruhat ordering of Coxeter groups. I was motivated especially by the seminal 1979 paper of D.A. Kazhdan and G. Lusztig [1], which has led to rapid progress in representation theory and which deserves to be regarded as a fundamental chapter in the theory of Coxeter groups.

Part I deals concretely with two of the most important types of Coxeter groups: finite (real) reflection groups and affine Weyl groups. The treatment is fairly traditional, including the classification of associated Coxeter graphs and the detailed study of polynomial invariants of finite reflection groups.

Part II is for the most part logically independent of Part I, but lacks motivation without it. Chapter 5 develops the general theory of Coxeter groups, with emphasis on the ‘root system’ (following Deodhar [4]), the Strong Exchange Condition of Verma, and the Bruhat ordering.