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 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 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 » 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.
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