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].
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