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 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.
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