CHAPTER 1
Introduction
In this paper we study triples (G, H, V ), where G is a simple algebraic group
over an algebraically closed field K, H is a closed positive-dimensional subgroup
of G and V is a rational irreducible KG-module such that V is irreducible as a
KH-module. We will refer to such a triple (G, H, V ) as an irreducible triple.
The study of irreducible triples has a long history, dating back to fundamental
work of Dynkin in the 1950s. In [7], Dynkin determined the maximal closed con-
nected subgroups of the classical matrix groups over C. One of the most difficult
parts of the analysis concerns irreducible simple subgroups; here Dynkin lists all the
triples (G, H, V ) where G is a simple closed irreducible subgroup of SL(V ), differ-
ent from SL(V ), Sp(V ), SO(V ), and H is a positive-dimensional closed connected
subgroup of G such that V is an irreducible module for H.
Not surprisingly, the analogous problem in the positive characteristic setting
is much more difficult. For example, complete reducibility may fail for rational
modules for simple groups, and there is no general formula for the dimensions of
irreducible modules. In the 1980s, Seitz [23] initiated the investigation of irreducible
triples over fields of positive characteristic as part of a wider study of the subgroup
structure of finite and algebraic simple groups. By introducing several new tools and
techniques, which differed greatly from those employed by Dynkin, Seitz determined
all irreducible triples (G, H, V ), where G is a simply connected simple algebraic
group of classical type, defined over any algebraically closed field K, and H is a
closed connected subgroup of G. In addition, Seitz also assumes that V = W, W
∗,
where W is the natural KG-module. This was extended by Testerman [26] to
exceptional algebraic groups G (of type E8, E7, E6, F4 or G2), again for H a
closed connected subgroup. In all cases H is semisimple, and in view of Steinberg’s
tensor product theorem, one may assume that V is p-restricted as a KG-module
(where p 0 denotes the characteristic of K, and one adopts the convention that
every KG-module is p-restricted if p = 0). In both papers, the irreducible triples
(G, H, V ) are presented in tables, giving the highest weights of the modules V |G
and V |H .
The work of Seitz and Testerman provides a complete classification of the triples
(G, H, V ) with H connected, so it is natural to consider the analogous problem for
disconnected positive-dimensional subgroups. A recent paper of Ghandour [11]
handles the case where G is exceptional, so let us assume G is a classical group
(of type An, Bn, Cn or Dn). In [8, 9], Ford studies irreducible triples in the spe-
cial case where G is a simple classical algebraic group over an algebraically closed
field of characteristic p 0, and H is a closed disconnected subgroup such that
the connected component
H0
is simple and the restriction V |H0 has p-restricted
1
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