10 2. PRELIMINARIES
a standard Frobenius morphism, and
ViF
i
is the KG-module obtained by preceding
the action of G on Vi by the endomorphism F
i.
By a slight abuse of terminology,
it is convenient to say that every dominant weight is p-restricted when p = 0.
Suppose G and H satisfy the conditions in Hypothesis 1. Write X = H0,
so X is a simple algebraic group of rank m, say. Fix a maximal torus TX of X
contained in T and let {δ1, . . . , δm} be the fundamental dominant weights for TX
corresponding to a choice of base Δ(X) = {β1, . . . , βm} for the root system Φ(X) of
X. By hypothesis (see condition S2), X acts irreducibly and tensor-indecomposably
on W ; let δ =

i
biδi denote the highest weight of W as a KX-module. Note that
δ is p-restricted (see condition S5).
In this paper we adopt the standard labelling of simple roots and fundamental
weights given in Bourbaki [3]. Finally, we will write Y = Φ to denote a semisimple
algebraic group Y with root system Φ.
2.2. Weights and multiplicities
Let V be an irreducible KG-module with p-restricted highest weight λ =

i
aiλi, that is, let V = VG(λ). Let Λ(V ) denote the set of weights of V and
let mV (μ) be the multiplicity of a weight μ Λ(V ), so mV (μ) is simply the dimen-
sion of the corresponding weight space Vμ. Recall that we can define a partial order
on the set of weights for T by the relation μ ν if and only if μ = ν
∑n
i=1
ciαi
for some non-negative integers ci. In this situation, if μ and ν are weights of V and
μ ν, then we say that μ is under ν. Note that if μ is a weight of V , then μ λ.
If J is a closed subgroup of G and TJ is a maximal torus of J contained in T ,
then we abuse notation by writing λ|J to denote the restriction of λ X(T ) to the
subtorus TJ . Let e(G) be the maximum of the squares of the ratios of the lengths
of the roots in Φ(G). Here we record some useful preliminary results on weights
and their multiplicities.
Lemma 2.2.1. If ai = 0 then μ = λ dαi Λ(V ) for all 1 d ai. Moreover,
mV (μ) = 1.
Proof. This follows from [26, 1.30].
Recall that a weight μ = λ

i
ciαi Λ(V ) is subdominant if μ is a dominant
weight, that is, μ =

i
diλi with di 0 for all i.
Lemma 2.2.2. Suppose μ is a weight of the Weyl module WG(λ),∑ and assume
p = 0 or p e(G). Then μ Λ(V ). In particular, if μ = λ
i
ciαi is a
subdominant weight then μ Λ(V ).
Proof. This follows from [22, Theorem 1].
Corollary 2.2.3. Suppose μ Λ(V ), and assume p = 0 or p e(G). Then
μ Λ(V ) for all α
Φ+(G)
and integers k in the range 0 k μ, α .
Proof. The set of weights of the Weyl module WG(λ) is saturated (see [14,
Section 13.4]), so the result follows from Lemma 2.2.2.
Previous Page Next Page