12

T. A. SPRINGER

these cocycles for the relation: (cs) ~ (c's) if and only if there is c e C such that

c's =

c~l

• cs • (sc), form the 1-cohomology set

Hl(k,

C). It has a privileged ele-

ment 1, coming from the constant function cs = e.

3.2. Reductive k-groups. Now let G be a connected reductive fc-group. It is said to

be quasi-split if it contains a Borel subgroup which is defined over k (this is a very

restrictive property). G is split (over /:) if it has a maximal torus which is defined

over k and /c-split. In this case G is quasi-split.

EXAMPLE.

G = 50(F) (see [2, pp. 15-16]). This is quasi-split but not split if and

only if the dimension n of the underlying vector space is even and the index equals

in - 1.

From the splitting of 2.14 one concludes that G is an inner form of a quasi-split

group.

Now let B be a Borel subgroup of G and T c B a maximal torus, both defined

over k5. Let cjjQ(G) = (X, A, Xv, Ay) be the based root datum defined by (G, B, T).

If s G rk there is gs e G(ks) such that

intfoX^O = B, int(gs)(sT)=T.

Then int(gs)o5- defines an automorphism of T depending only on s (since the coset

Tgs is uniquely determined). This automorphism determines an automorphism

fj,G(s) of X, permuting the elements of A (since int(gs) o s fixes B). It is easy to check

that juG defines a homomorphism juG: Tk - Aut (J0(G). Let G' be a £-form of G.

Then ^G = ^ if and only if G and G' are inner forms of each other.

3.3. Restriction of the base field [22, 1.3]. Let / c ks be a finite separable exten-

sion of A:. Let Gbean /-group. Then there exists a &-group H = i*//*G charac-

terized by the following property [2, 1.4]: for any /t-algebra A we have H(A) =

G(A ®k I). In particular, H(k) = G(/). Let 2 be the set of ^-isomorphisms / -* ks.

We then have H(ks) =

G(Ar5)2'.

The action of rk = Gal(£5/A:) on H(k5) is as follows.

If ^ e G(ksY is a function on 2 with values in (j(fc5), and s e rk, then, for a e 2",

(.y • $(j) = 0C? • a).

Rl/kG is obtained from G by restriction of the ground field from / to /:. If G is

connected or reductive then so is Rl/k G.

Now let G be connected and reductive. Fix B and T (defined over ks) as in 3.2

and let (poiG) be the based root datum defined by (G, B, T). Then H = i?//*G con-

tains the Borel subgroup Bx = J?- and the maximal torus Tx =

J2".

The based

root datum p0(Ri/kG) (relative to Bx and T{) is then

^(G)1.

The action of V^ on the

lattice Jf2, is like before: if s e rh (j e X2 then (s • fi)(a) = ^(^ • (7).

3.4. Anisotropic reductive groups. A connected reductive A:-group G is called

anisotropic (over /:) if it has no nontrivial /c-split /r-subtorus.

EXAMPLES,

(i) Let F be a nondegenerate quadratic form on a Avector space

(char k ^ 2). Let G = SO(F) be the special orthogonal group of F (the identity

component of the orthogonal group 0(F)). It is anisotropic over k if and only if F

does not represent 0 over k (the proof is given in [2, p. 13]).

(ii) If A: is a locally compact (nondiscrete) field then G is anisotropic over k if and

only if G(k) is compact.

(iii) If k is any field then G is anisotropic if and only if G(k) has no unipotent

elements 7* e and the group of its Arational characters Hom*(G, GL^ is trivial.