2.4. INVARIANT FORMS 11

Lemma 2.2.4. Suppose G = An and λ = aλi + bλj with i j and ab = 0. If

1 ≤ r ≤ i and j ≤ s ≤ n then μ = λ − (αr + · · · + αs) ∈ Λ(V ) and

mV (μ) =

j − i a + b + j − i ≡ 0 (mod p)

j − i + 1 otherwise.

Proof. This is [23, 8.6].

Lemma 2.2.5. Let W be equipped with a non-degenerate form f and let Y =

Cl(W ) = Isom(W ) be the corresponding simple group. Consider the natural em-

bedding Y SL(W ). Then there exists a choice of maximal tori TY Y and

T SL(W ), and a choice of Borel subgroups BY Y and B SL(W ) with

BY B, such that if {δ1, . . . , δ } and {λ1, . . . , λn} are the associated fundamen-

tal dominant weights, respectively, then λi|Y = δi for i ≤ − 2. In particular,

if λ =

∑n

i=1

aiλi is a dominant weight with aj = 0 for some j ≤ − 2, then

λ|Y =

∑

i=1

biδi, with bj = 0.

Proof. Let {β1, . . . , β } and {α1, . . . , αn} be bases of the respective root sys-

tems dual to the given weights. Consider the stabilizers in Y and SL(W ) of a max-

imal totally singular subspace of W (with respect to the form f). We may assume

that these correspond to the parabolic subgroups whose root systems have bases

{β1, . . . , β

−1

} and {α1, . . . , α

−1

, α

+1

, . . . , αn}, respectively, and that αi|Y = βi

for all 1 ≤ i ≤ − 1. In particular, we may assume that λ1|Y = δ1. Since

λi = iλ1 −

∑i−1

k=1

(i − k)αk for i 1 (see [14, Table 1]), the result follows.

2.3. Tensor products and reducibility

Suppose V1 and V2 are p-restricted irreducible KY -modules, where Y is a simply

connected simple algebraic group. Then according to [23, 1.6], the tensor product

V1 ⊗ V2 is an irreducible KY -module only under some very tight constraints. In

particular, if Y is simply laced (that is, if e(Y ) = 1) and at least one of the Vi is

nontrivial, then V1 ⊗ V2 is reducible (see [23, 1.6]).

Proposition 2.3.1. Let Y be a simple algebraic group and let V = VY (λ) be a

p-restricted irreducible KY -module. Then V can be expressed as a tensor product

V = V1 ⊗ V2 of two nontrivial p-restricted irreducible KY -modules Vi = VY (μi) if

and only if the following conditions hold:

(i) Y has type Bn,Cn,F4 or G2, with p = 2, 2, 2, 3, respectively.

(ii) λ = μ1 + μ2 and μ1 (respectively μ2) has support on the fundamental

dominant weights corresponding to short (respectively long) simple roots.

In particular, if Y is simple and e(Y ) = 1 then every p-restricted irreducible

KY -module is tensor indecomposable.

2.4. Invariant forms

Define G, H and W as in Hypothesis 1, and set X =

H0.

Let TX be a maximal

torus of X contained in T . Recall that W is self-dual as a KX-module, so X

fixes a non-degenerate bilinear form f on W . We state a result of Steinberg (see

[25, Lemma 79]), which determines the nature of this form. To state the result,