Introduction

This paper is a contribution to the study of the subgroup structure of simple algebraic

groups of exceptional type. The maximal closed connected subgroups of these groups

were determined in [Se2], subject to some mild restrictions on the characteristic p

of the underlying field. Here we take the study further, and investigate arbitrary

closed connected reductive subgroups X of an exceptional algebraic group G, again

with mild characteristic restrictions (in particular, p — 0 or p 7 covers all the

restrictions).

We obtain results which determine the embeddings of arbitrary closed connected

semisimple subgroups in G. We show that if X is such a subgroup, then X is em-

bedded in an explicit way in a "subsystem subgroup" of G - that is, a semisimple

subgroup which is normalized by a maximal torus of G. Subsystem subgroups are

constructed naturally from subsystems of the root system of G; this therefore de-

termines the embedding of X in G. As a consequence, when p — 0 there are only

finitely many conjugacy classes of such subgroups X, whereas there are infinitely

many when p 0. The connection with subsystem subgroups is useful in various

ways. For example, it is very helpful in finding centralizers of subgroups and in

restricting representations.

We present tables which give all the conjugacy classes of simple subgroups X of

G of rank at least 2, their connected centralizers, and their actions on £(G), the Lie

algebra of G. For subgroups X of type Ai, we associate with each such subgroup a

labelled Dynkin diagram, and prove that the conjugacy class of X is determined by

its labelled diagram.

Our proofs are based on Theorem 1, which states that if the reductive subgroup

X lies in a parabolic subgroup P = QL of G, with unipotent radical Q and Levi

subgroup i , then some conjugate of X lies in L. We also use this result to prove

that CG(X) is always reductive.

For simple subgroups X we establish that with essentially one exception, X is

determined up to (Aut G)-conjugacy by its composition factors on L(G); and that

if X is of rank at least 2, and p is a good prime for G, then

CUQ\{X)

—

L(CG(X)).

Some of our proofs require detailed information concerning the restrictions of

certain G-modules to various subgroups of G, such as maximal rank subgroups.

Many results of this type can be found in Section 2.

We now state our results in detail. Throughout, let G be a simple algebraic group

of exceptional type over an algebraically closed field K of characteristic p. In order

to specify our assumptions on p, we define, for certain simple subgroups X of G, an

integer N(X, G), as given in the following table.

Received by the editors April 13, 1994

Second author supported by an NSF grant and an SERC Visiting Fellowship