In this section we develop our basic machinery for showing that for a large number
of reductive groups X and irreducible X-modules V, the semidirect product XV has
just one conjugacy class of closed complements to V. As outlined in the Introduction,
this lies at the heart of the proof of Theorem 1. A key result is Proposition 1.5,
which says that with one specific exception, if there is more than one class of closed
complements then there is a rational indecomposable X-module which is an extension
of V by the trivial module. (This is of course elementary if we drop the words "closed"
and "rational".) We go on to apply this result to some particular examples of X and
Four preliminary lemmas are required to prove Proposition 1.5. The first two are
well known to experts.
Lemma 1.1 Let X be a classical connected simple algebraic group in characteristic
p, with natural module V — Wx(^i)- Regard X as a subgroup of GL(V), and L(X)
as an X-submodule of V ®V* (¥ gl(V)).
(i) If X — An, then L(X) is of codimension 1 in V ® V*, and Z(L(X)) has
(ii) Ifp ^ 2 and X = Bn or Dn, then L(X) = A V .
(Hi) Ifp ^ 2 and X = Cn, then L(X) =
Lemma 1.2 Let f : X -+ Y be a homomorphism of connected algebraic groups, with
X simple. Assume that the differential d(j) — 0. Then (f) can be factored through a
Frobenius morphism of X.
Proof. We may suppose that cf is surjective, and view both X and Y as Chevalley
groups. Let K be the underlying algebraically closed field, of characteristic p. If U is
a root subgroup of A, then U is isomorphic to
with coordinate ring K[x\. Sim-
ilarly, the image has coordinate ring K[y]. Using the fact that f is a homomorphism
and df) = 0, one easily checks that (f*(y) =
for some power q of p. A similar
conclusion holds when f is restricted to a 1-dimensional torus of X.
Fix a maximal torus T of X, and a corresponding system of root groups. Using
the above remarks, together with the connectivity of the Dynkin diagram of X and
the commutator relations, we see that for each root a in the root system of A", there
is a power qa of p and an element ka £ K* such that 4(Ua(c)) —
a r e r o
°t subgroups of A", y , respectively. Conjugating by a suitable element
of j)(T) we may re-parametrize root groups so that ka — 1 for all fundamental roots
a. From the usual expressions for ha(c) as products of elements of Ua and U-a, we
find that &_a = 1 for each fundamental root a.