(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
if n 2).
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