1. INTRODUCTION 7

G H W |H0 k Conditions

An Bl ω1 1 k 2l n = 2l, p = 2

An Dl.2 ω1 1 k 2l − 1 n = 2l − 1, p = 2

An NG(Tn) − 1 k n

An Al

2.2

ω1 ⊗ ω1 2, n − 1 n + 1 = (l +

1)2,

p = 2

An Al ω2 2, n − 1 n =

(l2

+ l − 2)/2, l ≥ 3, p = 2

An Al 2ω1 2, n − 1 n = (l2 + 3l)/2, p = 2

A26 E6 ω1 2, 3, 4, 23, 24, 25 p = 2, (p = 2, 3 if k = 3, 4, 23, 24)

A15 D5 ω5 2, 3, 13, 14 p = 2, (k, p) = (3, 3), (13, 3)

Cn Dn.2 ω1 1 k n p = 2

C28 E7 ω7 2 p = 2

3, 4, 5 p = 3, (k, p) = (5, 5)

C16 D6 ω6 2, 3 p = 2, (k, p) = (3, 3)

C10 A5.2 ω3 2, 3 p = 2

C7 C3 ω3 2, 3 p = 2, (k, p) = (2, 3), (3, 7)

C4 C1

3.S3

ω1 ⊗ ω1 ⊗ ω1 2, 3 p = 2, (k, p) = (3, 3)

C3 C1

3.S3

ω1 ⊕ ω1 ⊕ ω1 2 p = 3

C3 G2 ω1 2 p = 2

D4 C1

3.S3

ω1 ⊗ ω1 ⊗ ω1 3 p = 2

Table 3. H irreducible on all KG-composition factors of

Λk(W

)

G H W |H0 k Conditions

An Cl ω1 all n = 2l − 1

B3 G2 ω1 2 p = 2

B3 A2.2 ω1 + ω2 2 p = 3

B6 C3 ω2 2 p = 3

B12 F4 ω4 2 p = 3

Table 4. H irreducible on all KG-composition factors of

Sk(W

)

the highest weight λ of V . For example, consider the generic case H = X t , where

t is an involutory graph automorphism of X. If we take PX to be a t-stable Borel

subgroup of X then we find that the Levi factor L has an A1 factor, and the restric-

tion of λ to a suitable maximal torus of L affords the natural 2-dimensional module

for this A1 factor (see Lemma 2.8.2). By combining this observation with our lower

bounds on the dimensions of the quotients arising in (1.1), we can obtain severe

restrictions on the highest weight of W (viewed as an irreducible KX-module). As

noted in Lemma 2.8.2, even if our analysis of the quotients in (1.1) does not rule

out the existence of an A1 factor of L , we still obtain very useful restrictions on the

coeﬃcients ai when we express λ =

∑

n

i=1

aiλi as a linear combination of fundamen-

tal dominant weights. Further restrictions on the ai can be obtained by considering