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 .