1. INTRODUCTION 3
an explicit description of the embeddings of the subgroups in the family S is not
available, so in this situation an entirely different approach is required.
The main aim of this paper is to determine the irreducible triples (G, H, V )
in the case where H is a positive-dimensional disconnected subgroup in the col-
lection S. More precisely, we will assume (G, H, V ) satisfies the precise conditions
recorded in Hypothesis 1 below. By definition (see [18, Theorem 1]), every positive-
dimensional subgroup H S has the following three properties:
S1. H0 is a simple algebraic group, H0 = G;
S2.
H0
acts irreducibly and tensor indecomposably on W ;
S3. If G = SL(W ), then H0 does not fix a non-degenerate form on W .
In particular, since we are assuming H is disconnected and
H0 is irreducible,
it follows that H Aut(H0) modulo scalars, that is,
HZ(G)/Z(G)
Aut(H0Z(G)/Z(G)),
so
H {Am.2, Dm.2,D4.3,D4.S3,E6.2}.
(Here
Aut(H0)
denotes the group of algebraic automorphisms of
H0,
rather than
the abstract automorphism group.) We are interested in the case where H is max-
imal in G but we will not invoke this condition, in general. However, there is one
situation where we do assume maximality. Indeed, if G = Sp(W ), p = 2 and H fixes
a non-degenerate quadratic form on W then H GO(W ) G. Since the special
case (G, H) = (Cn,Dn.2) with p = 2 is handled in [6] (see [6, Lemma 3.2.7]), this
leads naturally to the following additional hypothesis:
S4. If G = Sp(W ) and p = 2, then H0 does not fix a non-degenerate quadratic
form on W .
Note that if H0 is a classical group, then conditions S3 and S4 imply that
W is not the natural KH0-module. Finally, since W is a tensor indecomposable
irreducible KH0-module, we will assume:
S5. W is a p-restricted irreducible KH0-module.
Our methods apply in a slightly more general setup; namely, we take H
GL(W ) with H0 G GL(W ) satisfying S1 S5. To summarise, our main aim
is to determine the triples (G, H, V ) satisfying the conditions given in Hypothesis
1, below.
Remark 1. As previously noted, if p = 0 we adopt the convention that all
irreducible KG-modules are p-restricted. In addition, to ensure that the weight
lattice of the underlying root system Φ of G coincides with the character group of
a maximal torus of G, in Hypothesis 1 we replace G by a simply connected cover
with the same root system Φ.
Hypothesis 1. G is a simply connected cover of a simple classical algebraic
group Cl(W ) defined over an algebraically closed field K of characteristic p 0,
H GL(W ) is a closed disconnected positive-dimensional subgroup of Aut(G) sat-
isfying S1 S5 above, and V is a rational tensor indecomposable p-restricted irre-
ducible KG-module with corresponding representation p : G GL(V ). In addition,
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