2

1. SOME THEORY OF LINEAR ALGEBRAIC GROUPS

Lie type. To mention a few celebrated examples of importance for us, the Borel-

Tits theorem, the Steinberg connectedness theorem and the high weight theory of

irreducible modules, all on the algebraic group level, have direct and fundamental

consequences for the structure of local subgroups and representations of the finite

groups of Lie type.

Accordingly, this first chapter gives a summary review of the theory of groups

of Lie type over the algebraic closure of a finite field, and is thus centered on

the theory of reductive linear algebraic groups over such a field. The descent via

Steinberg endomorphisms to finite groups of Lie type will be discussed in the next

chapter. Our purpose here is to state as rapidly as possible just what we need to get

directly to the questions on finite groups of Lie type which will be important in our

analysis of the finite simple groups. Consequently our approach distorts the theory

of algebraic groups in some ways: certain ideas fundamental for the development

of the theory, such as quotients, completeness, and Lie algebra of a group are

barely if ever mentioned. Nor are the results and ideas necessarily presented here

in an order natural to their logical development; for instance, quotient groups are

mentioned right away in Proposition 1.1.2b although their construction requires

considerable machinery and preparation. There are several excellent accounts of

a full development of the theory of algebraic groups; for example [Borl, Ch3,

Hum2, Spl].

We divide the chapter into a large number of sections. For the most part we just

state definitions and results; the final section discusses references to the literature.

Throughout this chapter, r is a prime and Fr the algebraic closure of the field

Fr of r elements, and X, H, etc., will always be algebraic

groups1.

For the most

part we write F for Fr.

1.1. Fundamental Notions

DEFINITION

1.1.1. The Zariski topology on GLn(F) is the topology defined

by the condition that the closed sets be the solution sets of systems of polynomial

equations in the matrix entries and the function d : A \-^ (det A)"1 for A G GLn(F).

An P-linear algebraic group (which we abbreviate to i^-algebraic group or

just algebraic group) is a closed subgroup K of GLn(F) for some n. The Zariski

topology on K is the topology inherited from that of GLn(F).

If K is an algebraic group, the affine algebra F[K] is the F-algebra of func-

tions K โ F under pointwise operations, generated by the matrix entries and the

function d. The elements of F[K] are called the polynomial functions on K.

A morphism fi : K โ iJ, or morphism of algebraic groups, is a group homo-

morphism fi such that for every polynomial function / on H, f o 0 is a polynomial

function on K.

We make several elementary remarks. First, a morphism of algebraic groups

is in particular a continuous mapping. Moreover, a mapping (j) : K โ ยป H be-

tween algebraic groups is an isomorphism of algebraic groups if and only if it is

an isomorphism of groups and both (j) and 0 _ 1 are morphisms of algebraic groups.

Furthermore, there is a category whose objects are the algebraic groups and whose

arrows are the morphisms of algebraic groups. In the case of possible confusion

1 In later chapters, when we use the bar convention for homomorphic images of finite groups,

the context should prevent confusion with this use of bars.