14 MARTIN W. LIEBECK AND GARY M. SEITZ

Proof. All the modules listed occur in the tables in [BW], from which the result is

immediate. •

The next lemma gives us another tool for obtaining a conclusion like that of 1.7.

L e m m a 1.13 Let X be a simple algebraic group and let V be a finite-dimensional

rational irreducible X-module. Suppose that X contains tori E\,...,Er satisfying

the following conditions:

1. each E{ is fixed-point-free on V — {0},

2. [E{,Ej] = 1 for all i,j,

3. X = (Cx(Ei) :lir).

Then the semidirect product XV contains just one class of closed complements to V.

In particular, this holds if Z(X) acts trivially on V.

Proo f (cf. [LS2,1.8]). Let Xo be a closed complement to V in XV. We claim that

Ei lies in a unique conjugate of X0 in XV. For Xo contains a torus F\ which is

a complement to V in E\V. Then E\ — F^ for some v £ V, so E\ Xfi. If also

Ei X£(we V), then F?w~' X0H FXV = Fx. Hence vw'1 G NV{F1) = 1 (since

F\ is fixed-point-free), so v — w, proving the claim.

Replacing XQ by a suitable ^-conjugate, we may therefore assume that E\

Xo. Since E\ is fixed-point-free, Cxv{E\) — CxQv{E\) X

0

. Hence for any i,

Ei Cx(Ei) X0. Then CX(EZ) X0, and so X = (Cx(El) : 1 i r) XQ.

Therefore Xo = X , giving the result. •

In the following proposition, we adopt the notation aft . . . to denote the irreducible

module Vx(a^i + bX2 + •••)•

Propositio n 1.14 Let X be one of the following simple algebraic groups in charac-

teristic p, and let V be one of the irreducible X-modules listed, where q denotes a

positive power of p:

_X F

10® 10M, 10® OlW, 10® 20M, 10® 0 2 ^ ,

2(P ' 10® l l W , 2 0 ® 10^),02® 10^), 11® 1 0 ^

11,13,01® 0 1 ^ , 0 1 ® 10^),

2 ^ ^ ) 01®02(^),10®0l( 9 ),02®01^ )

100® 100^), 100® 010W, 100® 0 0 1 ^ ,

3 ^ "' 010 ® 100^),001 ® 100(9)

B3 001 ® 001(9)

Then there is just one class of closed complements to V in XV, and the same holds

for XV* (where V* is the dual ofV).