16

T. A. SPRINGER

The smallest nontrivial special case is n = 3, k = F2. Here & is a graph with 14

vertices and 21 edges (drawn in [20, p. 210]).

(iii) Let char k # 2. Let V be a vector space over A: (in the sense of algebraic

geometry). Let F be a nondegenerate quadratic form on V which is defined over k.

With respect to a suitable basis of V{k) we have

F(xu •••,*„) = x ^ + *2*»-i + •" + *rr»-?+i + F0(xq+i, •••,

A:„_9),

where FQ is anisotropic over k (i.e., does not represent zero nontrivially). The index

q of /MS the dimension of the maximal isotopic subspaces of V(k).

Let (7 = SO(F) be the special orthogonal group of F. It is a connected semi-

simple A"-group. A maximal Asplit torus S in G is given by the matrices of the form

diag(/1? ••-,/,, I,---, l,ff\ •••,

t~l).

Then Z(S) is the direct product of S and the anisotropic Agroup SO(F0).

For a description of a minimal parabolic /:-subgroup and the determination of

the relative root system

k

0 we refer to [2, p. 16]. The latter is of type Bq if 2q # n

and of type Dq otherwise. If q [n/2] there are always subgroups Ua of dimen-

sion 1 (notations of 3.5).

A geometric description of parabolic A:-subgroups similar to the one for GL(«)

can be given. They are in this case the isotropy groups of isotropic A'-flags in V, i.e.,

flags all of whose subspaces are isotropic with respect to F.

4. Special fields. Let G be a A'-group. In this section we discuss some special fea-

tures for particular A\

4.1. R and C. If G is a C-group then G(C) has a canonical structure of complex Lie

groups. The latter is connected if and only if G is Zariski-connected (this can be

deduced from Bruhat's lemma, compare 4.2).

Now let k = R. Then G(R) is canonically a Lie group.

4.2.

LEMMA,

(i) G(R) is compact if and only if the identity component G° is a reduc-

tive anisotropic R-group;

(ii) G(R) has finitely many connected components.

(i) is easily established. As to (ii), it suffices to prove this if G is connected reduc-

tive. In that case one reduces the statement, via Bruhafs lemma, to the case that

G is either compact or a torus. In these cases the assertion is clear.

G(R) need not be connected if G is Zariski-connected, as one sees in simple cases

(e.g., G = GL(rt)).

If G is a C-group then the real Lie group RC/R(G)(R) (see 3.3) is that defined by

the complex Lie group G(C).

4.3. Finite fields. Let k = Fq and let k be an algebraic closure. F denotes the

Frobenius automorphism x *-* & of k/k. The basic result here is Lang's theorem

[6, p. 171].

4.4.

THEOREM.

If G is a connected k-group then g v~*

g~l(Fg)

is a surjective map of

G{k) onto itself

Using that G is an inner form of a quasi-split /r-group (see 3.1) one deduces that

a connected reductive k-group is quasi-split. A complete classification of simple

A--groups can then be given.