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
factors of X on V , and we use the fact that every weight space of an irreducible
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  (and Seitz  initially) we study certain parabolic
subgroups of G constructed in a natural way from parabolic subgroups of
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
= 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
of type Dm with m ≥ 5; the special case
= D4 is handled separately in Chapter
6. Finally, the case
= E6 is dealt with in Chapter 5, and the short proof of
Theorem 5 is given in Chapter 7.
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.