EXCEPTIONAL ALGEBRAIC GROUPS
15
Proof. Consider X = A2. When V or F * is 10 ® l l W or 11 ® 1 0 ^ , Z(X) acts
nontrivially, so the result holds. For the other cases we apply 1.13 with r = 2 and
Ei = { d i a g ( a , a , a ~ 2 ) : a G A'*}, £
2
= {diag(a;~ 2 ,a, a) : a G AT*}.
Next let X = 5
2
. If V is 11,13,01® 1 0 ^ , 10® 0 1 ^ , 01® 0 2 ^ or 02® 01W, then
Z(X) acts nontrivially. And if V = 01 ® 0l'
g
' , regard X as C2 = 5^4, and relative
to a symplectic basis for the usual 5p4-module, take r = 2 and
J5i = { d i a g ( a , a , a ~ 1 , a " 1 ) : a G ii*} E2 = {diag (a, a" 1 , a - 1 , a ) : a G K*}.
Then 1.13 applies.
For all cases with X = A3, we let r = 2 and take E\,E2 as above.
Finally, consider X = B3 and V = 001 ® 0 0 1 ^ . Write W for the 8-dimensional
irreducible X-module 001. Now X contains a subgroup D D3 preserving a decom-
position of W as a direct sum of two 4-spaces W\ and W2. Relative to a basis of W
containing bases of W\ and W2, D contains a subgroup
E\ = { d i a g ( a , a , a ~ 1 , a ~ 1 , a ~ 1 , a ~ 1 , a , a ) : a G K*}.
On the usual 7-dimensional module for X , this subgroup E\ acts as
{diag (a 2 , a - 2 , 1 , 1 , 1 , 1 , 1 ) : a G K*}, so Cx{E\) contains B2. Therefore, if we define
E2 { d i a g ( a , a
- 1
, a , a
- 1
, a
- 1
, a , a
- 1
, a ) : a G K*}
(relative to the above basis of W) , then (Cx(Ei), Cx(E2)) = X and hence 1.13 again
gives the result.
Propositio n 1.15 Suppose that either X B2jV = 10 ® 1 0 ^ and p 2, or
X = G2,V = 1 0 ® 1 0 ^ and p 3 (where q
pa1a
1). Then the conclusion of
1.14 holds.
Proof. First consider X = B2,V 10 ® 10^). Let Xo be a closed complement
to V in XV. If A is a subgroup A\ of X generated by a short root group and its
opposite, then A acts as SO3 on Vx(10), so has composition factors of high weights
2,0,0. Hence the nontrivial composition factors of A on V are 2 ® 2 ^ , 2 and 2 ^ .
By 1.8, none of these has a rational indecomposable extension by the trivial module,
so by 1.5, there is only one class of closed complements to V in AV. Therefore,
conjugating Xo by an element of V, we have A X D X
0
.
Now
Cxv(A)0
=
TIVQ,
where
VQ
is a 4-dimensional subspace of V and T\ is a
1-dimensional torus in X. Conjugating X by an element of Cxv(A), we may assume
that ATi X fl X
0
. Then X fl X
0
contains a maximal torus of X , so there is a
1-dimensional torus T{ in X fl Xo such that T[ lies in a fundamental A\ in X. Then
Cxv{T{)° =
ViAfT[,
where V\ is a 1-space in V and A! is a fundamental A\ in X .
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