10 MARTIN W. LIEBECK AND GARY M. SEITZ

(i) there is a rational indecomposable extension ofVx(X) by the trivial module;

(ii) p = 2, X = Cn and A = 2l\\ for some i (or 2lX2 if n — 2).

Proof. Let W be a rational indecomposable extension Vx(#A)/Vx(0), and let 0 :

X — » SL(W) be the associated morphism. The differential d(/ maps L(X) to a

nilpotent subalgebra of sl{W).

Suppose that dj) ^ 0. Then L(X) has a nontrivial nilpotent quotient. Except

when X — Cnj p = 2, it is easy to check that each root element of L(X) is a

commutator, so L(X) has no nontrivial nilpotent quotient. Thus X = Cn and p — 2.

The ideal / of L(X) generated by the short root elements and a Cartan subalgebra of

L(X) acts trivially on W; write L(X) = / + ( e i , . . . , e2n) and pick w £ W — Vx(qX).

As X is trivial on W/Vx(q\), it follows that {e\w,.. .,e2nw) is X-invariant. Hence

dim V^(A) 2n, which forces A = 2*Ai (or 2*A2 if n — 2), as in conclusion (ii).

Now suppose that d(f) = 0. Then by 1.2, 0 = ^ c y , where aqi is a g'-power

Frobenius morphism of X and tp is a morphism X — 5,X(iy) with d^? 7^ 0. Moreover

q' q, and xj) corresponds to a rational indecomposable extension Vx(-2rA)/Vx(0).

If qf q then as above, (ii) holds. And if q' ~ q then (i) holds. •

For the next result, let X = X\.. .X& be a commuting product of simple con-

nected algebraic groups X{ in characteristic p. For each i, let Vi be a nontrivial

rational irreducible finite-dimensional A^-module, and set V = Vi ® . . . ® Vjt, an

irreducible X-module.

L e m m a 1.4 With the above notation, if there is a rational indecomposable extension

ofV by the trivial module, then the same holds for each Vi.

Proof. Suppose W is such an extension of V. Now V j X{ is a homogenenous

direct sum of Xt--modules isomorphic to V{. Ii there is no rational indecomposable

extension of V{ by the trivial module, then W | X{ is completely reducible, and so

Cw(Xi) is a 1-dimensional space. But Cw(Xi) is X-invariant, contrary to the fact

that W is indecomposable. D

The next proposition is a key result. In it, we view the semidirect product XV

as an algebraic group.

Proposition 1.5 As above, let X = X\ . . . Xk, V — V\ ® . . . ® Vk, where for each i,

Vi = Vxiif^i)- If the semidirect product XV contains more than one conjugacy class

of closed complements to V, then for each i, one of the following holds:

(i) there is a rational indecomposable extension of Vi by the trivial module;

(ii) p = 2,X; = Cn and m — 2^X\ (or

2JA2

if n — 2).