EXCEPTIONAL ALGEBRAIC GROUPS

11

Proof. We recall some standard material. A 1-cocycle is a (rational) function

7 : X — V such that 7(^1X2) = 7(^1)^2 + 7(^2) for all xi,x2 G X. There is an

action of X on the additive group of 1-cocycles, given by

l9{x)

= ) W - 7 ( j )

for all g, x G X. For v € V there is a 1-coboundary v given by v(x) — v — vx for all

x G X. With the above action of X , we have v9 = log for v G F, # G X . And for a

1-cocycle 7 and # G X ,

7^ ~ 7 = -7(ff)»

so X acts trivially on the factor group of 1-cocycles modulo 1-coboundaries, namely

H\X,V).

Now suppose that Y is a closed complement to V in X V , and that Y is not

F-conjugate to X . Then

y = {x-y(x) : x G X }

where 7 : X —• y is a 1-cocycle. Write 2/ for the element xj(x) of y .

Form the iT-vector space W = V + (7), and extend the action of X on V to an

action on W by setting

73 = 7 - 7(a)

for a: G X . This action also extends to Y via the sequence Y —• X 7 — • X — G i ( W )

(where the first map is inclusion and the second is projection).

We claim that the representation Y — GL(W) is rational. The action of X on

V is rational, so it is clear from the above sequence that the action of Y on V is also

rational. Now consider the sequence Y — XV — V, where the first map is inclusion

and the second is projection. Both maps are morphisms, so the map x' — 7(2;) is

rational. The claim follows.

Suppose that Y fixes a vector in W — V. Then there exists v € V such that

(v + *y)xf — v -\-^j for all x' G Y. It follows that (v + 7)2 = v + 7 for all x G X , so by

definition of 7$,

7(2) = — (v — vx)

for all x G X . Hence 7 = — v, which means that X and y are conjugate by — v, a

contradiction. Therefore i y is indecomposable under the action of Y.

Consider again the map Y — X V — X , and write y = Y\ .. . Y&, where for

each i,Y{ is the preimage of X{. This gives a morphism \{ : Y{ — X; which is an

isomorphism of abstract groups. We then get a morphism Y{ — X{ —*• Gly(y), and

Lemma 1.4 shows that as a 5^-module, V{ has a rational indecomposable extension

W{ by the trivial module.

Suppose now that L(Y{) has no nonzero nilpotent ideal. Then L(Y{) fl L(V) — 0.

Since the differential of the projection map X{V —• X{ has kernel L(V), it follows

that dji is an isomorphism. Hence f{ is an isomorphism of algebraic groups by [Sp,