CHAPTER 1

Introduction

This article concerns the subgroup structure of semisimple algebraic groups over

algebraically closed fields. Let G be a simple algebraic group over an algebraically

closed field K of characteristic p. When p is 0 the maximal closed subgroups of

positive dimension in G were classified in a paper of Dynkin in 1952, [Dyn52]. Seitz

[Sei87] initiated the study in positive characteristic, and following this, Liebeck

and Seitz [LS04] completed the determination of the maximal closed subgroups of

positive dimension in the case where G is of exceptional type. The list is surprisingly

small: parabolic subgroups, reductive subgroups of maximal rank, and just a small

(finite) number of conjugacy classes of further reductive maximal subgroups.

For many reasons one wants to extend this work to study all reductive sub-

groups of simple algebraic groups (not just maximal ones). For example, a result

of Richardson [Ric77] characterises those varieties G/H which are aﬃne as being

precisely those for which

H◦

is a closed reductive subgroup of G. One might think

that finding the connected reductive subgroups

H◦

is simply a matter of working

down from the maximal subgroups, but one quickly realises that there is a major

obstacle to this—namely, dealing with reductive subgroups in parabolics. If H is a

reductive subgroup of a parabolic P = QL (Q the unipotent radical, L a Levi sub-

group), one can continue working down through the maximal subgroups of L if H is

contained in a conjugate of L, but, as we describe later, it is much less easy to find

those H which are not in any conjugate of L. This leads to the following definition

of a G-completely reducible subgroup, originally introduced by Serre [Ser98].

Let H be a subgroup of a reductive algebraic group G. Then H is said to be

G-completely reducible (G-cr) if, whenever it is contained in a parabolic subgroup

P of G, it is contained in a Levi subgroup L of P . It is said to be G-irreducible

(G-ir), when it is contained in no proper parabolic of G. It is said to be G-reducible

(G-red) if it is not G-ir. It is said to be G-indecomposable (G-ind) when it is not

contained in any proper Levi subgroup of G.

These definitions generalise the notions of a representation of a group φ : X →

GL(V ) being completely reducible, irreducible or indecomposable respectively since

in the case where G = GL(V ) the image of X, φX is a subgroup of GL(V ) which

is G-cr, G-ir or G-ind, resp.

An analogue of Maschke’s Theorem orginally due to Mostow [Mos56] tells us

that if the characteristic of the field K, p is 0 then all connected reductive subgroups

of G(K) are G-cr. (See [Jan03, II.5.6(6)] and [Jan04, Lemma 11.24].) Following

the usual paradigm, the obvious question is: can we ensure that a subgroup H of

G is G-cr if we know that p is in some sense big?

Indeed, for the simple algebraic groups, classical and exceptional, respectively,

McNinch [McN98] and Liebeck-Seitz [LS96] have very specific answers. We give

the coarsest versions of each of the main results in these papers. For G a classical

1