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,