0 - Basic results on algebraic groups

Summary

A finite group of Lie type will be defined in this book as the group of points over a finite field of a (usually connected) reductive algebraic group over the algebraic closure of that finite field. We begin by recalling the definition of these terms and the basic structure theory of reductive algebraic groups.

Let us first establish some notations and conventions we use throughout. If g is an automorphism of a set (resp. variety, group, …)X, we will denote by X^g the set of fixed points of g, and ^gx the image of the element x ∈ X by g. A group G acts naturally on itself by conjugation, and we will hence write ^gh for ghg^-1, where g and h are elements of G. We will write Z(G) for the centre of G; if X is any subset of G, we put N_G(X) = {g ∈ G | ^gX = X} and C_G(X) = {g ∈ G | gx = xg for all x ∈ X}.

We will consider only affine algebraic groups over an algebraically closed field k (which will be taken to be F_q from chapter 3 onwards), i.e., affine algebraic varieties endowed with a group structure such that the multiplication and inverse maps are algebraic (which corresponds to a coalgebra structure on the algebra of functions on the variety). For such a group G, we will call elements of G the elements of the set G(k) of k-valued points of G. We generally will use bold letters for algebraic groups and varieties.