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

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