REDUCTIVE GROUPS

11

We have a decomposition G = G' • 5, where G' is the derived group of G (which

is semisimple) and where S is a torus, viz. the identity component of the center C

of G. We have T = 7" • S, where I" is a maximal torus of G'.

We use the notations of 1.1 and 1.7.

The following facts can be checked without difficulty.

(a) 0(G', T) = (X/X0i 0, Gv, 0V) (we view $ as a subset ofXjX^ as we may by

1.2).

(b) The character group Homalg.gr. (C, 0*) is X/Q, and we have Z*(5) = X/Q,

XV(S) = X%.

(c) The isogeny G' x S - G defines the canonical isogeny of (G, T) (see 1.7).

This fact was already used in the proof of 2.5(H).

It follows that G is semisimple if and only if 0(G, T) is semisimple. In that case

we say that G is adjoint if X = Q and simply connected if X = P (notation of 1.8).

From 2.5(H), using what was said in 1.8, we see that a semisimple group G is adjoint

(resp. simply connected) if and only if a central isogeny j)\ G - G' (resp. fi: G' -»

G) is an isomorphism.

In the case of a general reductive G we have the following facts.

(d) The derived group G' is adjoint oX = Q ® X0oX^ = pv @ xtf.

(e) G' is simply connected o P c X + (X0 ® Q) =

(?v

= g

v

.

(f) The center of G is connected o Q = Q o P

v

c Z

v

+ (Xtf ® g).

3. Reductive groups (relative theory). Here we let a ground field k c Q come into

play. We denote by k the algebraic closure of k in £ and by ks its separable closure.

A linear algebraic group G which is defined over k will be called a k-group. We

then denote by G(k) the group of its ^-rational points (and not by Gk, as in [2]). If

A is a fc-algebra, we denote by G(A) the group Hom*(£[G], ^4) (see [2]).

3.1. Forms of algebraic groups [16, III, §1]. Let G and G' be ^-groups. G' is said

to be a k-form of G if G and G' are isomorphic over Q.

EXAMPLE,

k = R. Then J7(«) is an R-form of GL(«).

To describe Aforms one proceeds as follows. The fc-group G is completely de-

termined by the group G(ks) of fc5-rational points. This means the following: if

G - GL(n) is an isomorphism of G onto a closed subgroup of GL(«), everything

being defined over k, then the subgroup G(ks) of GL(«,fc5)determines G, up to A:-iso-

morphism. The fact that G is defined over k is reflected in an action of the Galois

group Fk = Gal(k$/k) on G(ks). The A:-forms G' of G can be described as follows

(up to ^-isomorphism). We have G'(ks) = G(ks) and there is a continuous func-

tion c:s »- c5 of /"* to the group of /:5-automorphisms of G (the Galois group being

provided with the Krull topology and the second group with the discrete topol-

ogy), satisfying

(*) cst = c5 • s(ct) (s, t e Fk),

such that the action of Fk on G'(ks) (denoted by (s, g) «- s * g) is obtained by

"twisting" the original action with c: (s*g) = cs(s-g). G' is ^-isomorphic to G

if and only if there exists an automorphism c such that cs =

c~l

-sc.

We say that G' is an inner form of G if all cs are inner automorphisms.

If C is a group on which Fk acts, the continuous functions s *-+ cs ofFk to C which

satisfy (*) are called 1-cocycles of Fk with values in C. The equivalence classes of