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,
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