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

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