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,
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