2 1. INTRODUCTION

composition factors. These extra assumptions help to simplify the analysis. Never-

theless, under these hypotheses Ford discovered a very interesting family of triples

(G, H, V ) with G = Bn and H = Dn.2 (see [8, Section 3]). Furthermore, these

examples were found to have applications to the representation theory of the sym-

metric groups, and led to a proof of the Mullineux conjecture (see [10]). However,

for future applications it is desirable to study the general problem for classical

groups, without any extra conditions on the structure of H0 and the composition

factors of V |H0 .

In this paper we treat the case of irreducible triples (G, H, V ) where G is of clas-

sical type, H is maximal among closed positive-dimensional subgroups of G and V

is a p-restricted irreducible tensor indecomposable KG-module. Using Steinberg’s

tensor product theorem, one can obtain all irreducible triples (G, H, V ) as above,

without requiring the p-restricted condition on V . We now explain precisely the

content of this paper.

Let G be a simple classical algebraic group over an algebraically closed field K

of characteristic p ≥ 0 with natural module W . More precisely, let G = Isom(W ) ,

where Isom(W ) is the full isometry group of a suitable form f on W , namely, the

zero bilinear form, a symplectic form, or a non-degenerate quadratic form. We

write G = Cl(W ) to denote the respective simple classical groups SL(W ), Sp(W )

and SO(W ) defined in this way. Note that G = Isom(W ) ∩ SL(W ), with the

exception that if p = 2, f is quadratic and dim W is even, then G has index 2 in

Isom(W ) ∩ SL(W ).

A key theorem on the subgroup structure of G is due to Liebeck and Seitz,

which provides an algebraic group analogue of Aschbacher’s well known subgroup

structure theorem for finite classical groups. In [18], six natural (or geometric)

families of subgroups of G are defined in terms of the underlying geometry of W ,

labelled Ci for 1 ≤ i ≤ 6. For instance, these collections include the stabilizers of

suitable subspaces of W , and the stabilizers of appropriate direct sum and tensor

product decompositions of W . The main theorem of [18] states that if H is a

positive-dimensional closed subgroup of G, then either H is contained in a subgroup

in one of the Ci collections, or roughly speaking,

H0

is simple (modulo scalars) and

H0

acts irreducibly on W . (More precisely, modulo scalars, H is almost simple in

the sense that it is contained in the group of algebraic automorphisms of the simple

group

H0.)

We write S to denote this additional collection of ‘non-geometric’

subgroups of G.

Let H be a maximal closed positive-dimensional disconnected subgroup of a

simple classical algebraic group G = Cl(W ) as above, let V be a rational irreducible

KG-module and assume V = W or W ∗, where W ∗ denotes the dual of W . The

irreducible triples (G, H, V ) such that V |H0 is irreducible are easily deduced from

the aforementioned work of Seitz [23], so we focus on the situation where V |H

is irreducible, but V |H0 is reducible. By Clifford theory, the highest weights of

the KH0-composition factors of V are H-conjugate and we can exploit this to

restrict the possibilities for V . This approach, based on a combinatorial analysis

of weights, is effective when H belongs to one of the geometric Ci families since

we have an explicit description of the embedding of H in G. In this way, all

the irreducible triples (G, H, V ) where H is a disconnected positive-dimensional

maximal geometric subgroup of G have been determined in [6]. However, in general,