8 1. INTRODUCTION

the flag (1.1) with respect to different parabolic subgroups PX = QX LX of X. This

is how we proceed.

This general set-up, based on the embeddings PX P , can also be found

in Ford’s work. The main difference comes in the consideration of the second

commutator quotient. In [23, 2.14(vi)], Seitz establishes an upper bound for the

dimension of [V, Q]/[[V, Q],Q] in terms of the dimensions of V/[V, QX ] and a certain

quotient of QX . However, this upper bound is only valid if the highest weights of

the composition factors of V |X are p-restricted.

To replace this technique, we carefully analyse the action of X on W in order

to obtain information on the restrictions of weights and roots for G to a suitable

maximal torus of X. In addition, we consider the action of certain

(Am)t

Levi

factors of X on V , and we use the fact that every weight space of an irreducible

K(A1)t-module

is 1-dimensional. Various considerations such as these enable us to

reduce to a very short list of possibilities for the highest weight λ of V .

Finally, some comments on the organisation of this paper. In Chapter 2 we

present several preliminary results which will be needed in the proof of our main

theorems. In particular, we recall some standard results on weights and their multi-

plicities, and following Ford [8] (and Seitz [23] initially) we study certain parabolic

subgroups of G constructed in a natural way from parabolic subgroups of

H0;

these

parabolic embeddings play a crucial role in our analysis. The remainder of the pa-

per is dedicated to the proof of Theorems 1 – 5 (with the focus on Theorem 3).

In Chapter 3 we assume

H0

= Am; the low rank cases m = 2, 3 require special

attention, and they are dealt with in Sections 3.4 and 3.5, respectively, while the

general situation is considered in Section 3.6. Next, in Chapter 4 we assume

H0

is

of type Dm with m ≥ 5; the special case

H0

= D4 is handled separately in Chapter

6. Finally, the case

H0

= E6 is dealt with in Chapter 5, and the short proof of

Theorem 5 is given in Chapter 7.

Acknowledgments

The first author was supported by EPSRC grant EP/I019545/1, and he thanks

the Section de Math´ ematiques at EPFL for their generous hospitality. The third

and fourth authors were supported by the Fonds National Suisse de la Recherche

Scientifique, grant number 200021-122267.