CHAPTER 2
Preliminaries
2.1. Notation and terminology
Let us begin by introducing some notation that will be used throughout the pa-
per. Further notation will be introduced in the subsequent sections of this chapter,
and we refer the reader to p.107 for a convenient summary of the main notation we
will use.
As in Hypothesis 1, let G be a simply connected cover of a simple classical
algebraic group Cl(W ) = SL(W ), Sp(W ) or SO(W ), defined over an algebraically
closed field K of characteristic p 0. Here Cl(W ) = Isom(W ) , where Isom(W ) is
the full isometry group of a form f on W , which is either the zero bilinear form,
a symplectic form or a non-degenerate quadratic form. It is convenient to adopt
the familiar Lie notation An, Bn, Cn and Dn to denote the various possibilities
for G, where n denotes the rank of G. Note that if G = SO(W ), p = 2 and W is
odd-dimensional, then G acts reducibly on W , so for the purpose of proving our
main theorems, we may assume that p = 2 if G is of type Bn. Also note that we
will often refer to G as a ‘classical group’, by which we mean the simply connected
version of G.
Fix a Borel subgroup B = UT of G, where T is a maximal torus of G and U
is the unipotent radical of B. Let Δ(G) = {α1, . . . , αn} be a corresponding base of
the root system Φ(G) =
Φ+(G)∪Φ−(G)
of G, where
Φ+(G)
and
Φ−(G)
denote the
positive and negative roots of G, respectively. We extend the notation Δ, Φ,
Φ+
and
Φ−
to any reductive algebraic group. Let X(T )

=
Zn
denote the character group of
T and let {λ1, . . . , λn} be the fundamental dominant weights for T corresponding
to our choice of base Δ(G). Here λi,αj = δi,j for all i and j, where
λ, α = 2
(λ, α)
(α, α)
with (,) the usual inner product on X(T )R = X(T ) ⊗Z R, and δi,j is the familiar
Kronecker delta.
There is a bijection between the set of dominant weights of G and the set of
isomorphism classes of irreducible KG-modules; if λ is a dominant weight then
we use VG(λ) to denote the unique irreducible KG-module with highest weight
λ, while WG(λ) denotes the corresponding Weyl module (recall that WG(λ) has a
unique maximal submodule M0 such that WG(λ)/M0

= VG(λ), and M0 is trivial
if p = 0). We also recall that if p 0 then a dominant weight λ =

i
aiλi is
said to be p-restricted if ai p for all i. By Steinberg’s tensor product theorem,
every irreducible KG-module decomposes in a unique way as a tensor product
V0
⊗V1F
⊗· ·
·⊗VrF
r
, where Vi is a p-restricted irreducible KG-module, F : G G is
9
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