12 MARTIN W. LIEBECK AND GARY M. SEITZ

4.3.4]. Now the map X{ —• Y{ -+ GL(Wi) (where the first map is (f)i

x)

shows that

conclusion (i) holds.

Thus we may assume that L(Y{) has a nonzero nilpotent ideal I which is the

kernel of d(fi. It follows that Y{ — Bn, p — 2 and / is the ideal generated by short

root elements of L(Y{)] and the image of L(Y{) under dfa is an ideal in £(X

Z

), whence

Xi — Cn. We claim that Y{ is of adjoint type. For if not, then Y{ is simply connected

and the ideal generated by short root elements of L(Y{) contains Z{L{Yi)) (since

Z(L(Yi)) lies in the Lie algebra of a short SL2 in Y{). But Z(L(Yi)) is not nilpotent,

which is a contradiction.

Hence Y{ is adjoint, and so there is a surjective morphism r : X{ -+ Y{. Then rfi

is a Frobenius morphism aq of X;, so it follows that there is a rational indecomposable

extension of the X^-module V^ by the trivial module. The result now follows from

1.3. •

The following standard result enables us to use Proposition 1.5 to show that for

many of the semidirect products XV as in the hypothesis (of 1.5), there is just one

class of closed complements to V. Recall that Wx(X) denotes the Weyl module for

X with high weight A, and Ext^(F , K) denotes the group of equivalence classes of

rational extensions of V by the trivial X-module K.

Proposition 1.6 [Ja, p.207] Let X be a simple algebraic group over K, and A a

dominant weight. Then

Extx(Vx(\),K) 2 Homx(rad(Wx(X)),K).

Corollary 1.7 Let X be a simple algebraic group over K of characteristic p, and

let V = Vy(A), where if(X,p) = (C

n

,2) then A / 2iX1 (or 2jX2 if n = 2). Suppose

that Homx(rad(Wx(X)), K) = 0. Then the semidirect product XV contains just one

class of closed complements to V.

Proof. This is immediate from 1.5 and 1.6. •

In the next few results, we present some information on Weyl modules which is

designed for the application of 1.5 and 1.7. Throughout, X is a simple algebraic

group in characteristic p.

In the first proposition, for a positive integer r we label by Vx(r) the irreducible

module for X — Ai with high weight r\i.

Proposition 1.8 Let X — A\, and suppose that there is a rational indecomposable

extension ofVx{r) by the trivial module. Then p 0 and Vx(r) is a Frobenius twist

of the module

Vx(p-2)®VX(1){P).

Proof. This follows from [AJL, 3.9]. •