We review invariants of reflection groups and reflection cosets up to giving a formula for the order of a finite reductive group. We then review the Springer correspondence between local systems on unipotent classes and characters of the Weyl group, and use it to describe the Lusztig–Shoji algorithm to compute Green functions.

We explain how the decomposition of unipotent Deligne–Lusztig characters is governed by the component groups of special unipotent classes. For each such class a Lusztig Fourier transform matrix is built from the Drinfeld double of the Lusztig quotient of the group of components.

This chapter defines Frobenius endomorphisms and Frobenius roots and gives the Lang–Steinberg theorem and its consequences using Galois cohomology. Then it studies and classifies finite reductive groups and studies their (B,N)-pairs.

This chapter studies endomorphism algebras of the Harish–Chandra induced representation from a cuspidal pair, at first on an arbitrary field, then in characteristic 0. This algebra is viewed as an Iwahori–Hecke algebra, allowing definition of Schur elements and generic degrees. The chapter ends with the character table of an Iwahori–Hecke algebra of type G2 in the generic case.

This chapter expounds the Harish–Chandra theory using the bimodule viewpoint to define induction and restriction. The Mackey formula is proven, leading to the definition of the Harish–Chandra series.

The theory of (B,N)-pairs is expounded and applied to reductive groups with its consequences on parabolic subgroups, Levi subgroups and centralisers of semi-simple elements.

We had two main aims in writing this edition:

• Be more self-contained where possible. For instance, we have added brief overviews of Coxeter groups and root systems, and given some more details about the theory of algebraic groups.

• While retaining the same level of exposition as in the first edition, we have given a more complete account of the representation theory of finite groups of Lie type.

In view of the second aim, we have added the following topics to our exposition:

• We cover Ree and Suzuki groups extending our exposition of Frobenius morphisms to the case more general of Frobenius roots.

• We have added to Harish-Chandra theory the topic of Hecke algebras and given as many results as we could easily do for fields of arbitrary characteristic prime to q, in view of applications to modular representations.

• We have added a chapter on the computation of Green functions, with a brief review of invariant theory of reflection groups, and a chapter on the decomposition of unipotent Deligne–Lusztig characters.

We associate a sign to F-stable reductive subgroups of a reductive group with a Frobenius root F, and define Curtis–Alvis duality. We prove its properties, in particular commutation with Harish–Chandra induction and to restriction to centralisers of semi-simple elements. We define the Steinberg character as the dual of the identity and use it to compute the number of unipotent elements.

We state properties of ℓ-adic cohomology and prove the corresponding properties of Lefschetz numbers.

We prove the existence of regular semi-simple elements, then of regular unipotent elements. We give properties of Gelfand–Graev representations and their duals. We compute their values on regular unipotent elements using Gauss sums. We decompose Gelfand–Graev representations into sums of regular characters and use them to prove the disjunction of Deligne–Lusztig characters corresponding to distinct rational semi-simple classes. The chapter ends with the character table of SL2.

We show how geometric conjugacy can be viewed in terms of the dual group. We explain the centre of a semi-simple group in terms of the affine root system. We explain Lusztig’s Jordan decomposition of characters. We express the characters of the general linear and unitary groups in term of Deligne–Lusztig characters and give the character table of GL2.

We describe the basic properties of algebraic groups, starting with tori and diagonalisable groups, then going on to solvable groups and Borel and parabolic subgroups, then going on to semi-simple and reductive groups, radical and unipotent radical, and finally giving as examples the classical linear, symplectic and orthogonal groups.

This chapter contains a review from scratch of the theory of Coxeter groups and root systems, then gives the main theorems on the structure of reductive groups and the isogeny theorem.

We give character formulae for Lusztig functors and deduce some consequences. We define uniform class functions. We prove that the regular, identity and Steinberg characters, as well as the characteristic function of a semi-simple class, are uniform and we express them as linear combinations of Deligne–Lusztig characters. We give the number of F-stable maximal tori. We give the degree of the Lusztig induction of a character.

These notes follow a course given at the Paris VII university during the spring semester of academic year 1987–88. Their purpose is to expound basic results in the representation theory of finite groups of Lie type (a precise definition of this concept will be given in the chapter “Rationality, Frobenius”).

We define Lusztig induction and restriction functors and give an approach towards the Mackey formula for these functors which is successful if one of the Levi subgroups involved is a torus. We give consequences for Deligne–Lusztig characters.