CHAPTER 1

Some Theory of Linear Algebraic Groups

There are several paths into the theory of groups of Lie type. Most of the

groups of Lie type are classical groups, arising from the stabilizers in GL(V) of

certain bilinear or quadratic forms / on the finite-dimensional vector space V:

(V, f) -• isometry group of (V, / ) .

Each simple classical group is then a large subquotient of such a stabilizer [Ar2,

D l , Dil, Tal], and the geometry of V with respect to the bilinear form sheds light

on the structure of the group, for example through Witt's Lemma (2.7.1 below).

The nonclassical ("exceptional") groups of Lie type can also be studied by means

of related geometries or as stabilizers of certain multilinear forms on vector spaces

V (e.g., [Til, A21]); such study normally proceeds on a group-by-group basis.

A second path begins with the finite-dimensional simple Lie algebras. For any

field F and finite-dimensional simple Lie algebra £ over the complex field C, there is

an untwisted group &(F) of Lie type—actually several closely related groups, each

arising from a different finite-dimensional module M for £. A uniform construction

for these groups can be given [Ch2, Stl] starting with the theory of complex Lie

algebras and their modules:

£ , M , F -• £(F) GL(M0®F).

Here Mo is an additive subgroup of M generated by a complex basis of M. The

twisted groups then arise as fixed point subgroups of certain automorphisms of the

untwisted groups. Such constructions are part of our assumed background [Cal,

Stl], as is the analysis from this perspective of various structural properties of

these groups: their automorphisms, BiV-structure, and parabolic subgroups.

Over an algebraically closed field the groups £(F) are linear algebraic groups,

and the extra variety structure which they carry makes possible extraordinarily

powerful tools for their analysis; most noteworthy is the classification theory [Ch3]

showing that for an algebraically closed field, the semisimple algebraic groups are

precisely the groups of Lie type. These tools in turn yield a third approach to the

finite groups of Lie type, twisted or untwisted. Indeed all the finite groups of Lie

type arise in a uniform way from simple algebraic groups over the algebraic closure

F of a finite field—as fixed point subgroups of certain endomorphisms a which we

shall call "Steinberg" endomorphisms [St2]:

1(F),* - CL{T)(a).

It turns out to be possible to pull considerable information down from the alge-

braic groups to the finite groups, and this approach has been and continues to be

indispensable for a thorough understanding of many aspects of the finite groups of

1

http://dx.doi.org/10.1090/surv/040.3/01