This volume gives an account of the representation theory of reductive algebraic groups over algebraically closed fields and over finite fields. It contains carefully coordinated chapters written by 9 leading workers in the area of algebraic groups.

The volume begins with an article by R.W. Carter introducing the basic concepts in the theory of linear algebraic groups. This includes the properties of well known subgroups such as maximal tori, Borel subgroups and parabolic subgroups, and a description of the classification of the simple algebraic groups by means of root systems and Dynkin diagrams.

There is a class of abstract groups, the Coxeter groups, which play a key role in the theory of algebraic groups. An article by R. Rouquier discusses the properties of Coxeter groups in general, and also the particular Coxeter groups such as Weyl groups and affine Weyl groups which appear in the theory of algebraic groups.

Various concepts from homological algebra are frequently used in the representation theory of algebraic groups. A chapter by B. Keller introduces these concepts, including abelian categories, derived categories and triangulated categories.

Finite reductive groups are defined as fixed point sets of reductive algebraic group under a Frobenius map. The representation theory in characteristic 0 of these groups was developed by Deligne and Lusztig. An article by M. Geek explains the basic properties of Frobenius maps and expounds the Deligne-Lusztig theory, including a parametrization of all irreducible representations of finite reductive groups.