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,
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