1. INTRODUCTION 5

G H λ δ λ|H0 κ Conditions

C10 A5.2 λ3 δ3 δ1 + 2δ4 or 2δ2 + δ5 2 p = 2, 3

C10 A5.2 λ2 δ3 δ2 + δ4 1 p = 2

B3 A2.2 2λ1 δ1 + δ2 2δ1 + 2δ2 1 p = 3

D7 A3.2 λ6, λ7 δ1 + δ3 δ1 + δ2 + δ3 1 p = 2

D13 D4.Y λ12, λ13 δ2 δ1 + δ2 + δ3 + δ4 1 p = 2, 1 = Y S3

Table 1. The triples (G, H, V ) with H G satisfying Hypothesis 2

H G. Moreover, for the case appearing in the final row we can take

H = D4.2,D4.3 or D4.S3.

Theorem 3. A triple (G, H, V ) satisfies Hypothesis 2 if and only if one of the

following holds:

(i) H G and (G, H, V ) is one of the cases in Table 1; or

(ii) H G and (G, H, λ) = (D10,A3.2,λ9 or λ10), δ = 2δ2, p = 2, 3, 5, 7 and

λ|H0 = 3δ1 + δ2 + δ3 or δ1 + δ2 + 3δ3.

In particular, the triples satisfying Hypothesis 1 are listed in Table 2.

G H λ δ λ|H0 κ Conditions

C10 A5.2 λ3 δ3 δ1 + 2δ4 or 2δ2 + δ5 2 p = 2, 3

D10 A3.2 λ9,λ10 2δ2 3δ1 + δ2 + δ3 or δ1 + δ2 + 3δ3 2 p = 2, 3, 5, 7

Table 2. The triples (G, H, V ) satisfying Hypothesis 1

Remark 6. Note that in case (ii) of Theorem 3, we have A3.2 D10.2, but the

graph automorphism of A3 is not contained in the simple group D10 (see Lemma

3.5.7). This is the only triple (G, H, V ) satisfying Hypothesis 2 with H G.

By combining Theorem 2 with the main theorems of [6] and [23], we obtain

the following result.

Theorem 4. Let G be a simple classical algebraic group over an algebraically

closed field K of characteristic p ≥ 0, H a maximal positive-dimensional closed

subgroup of G, and let V be a tensor indecomposable p-restricted irreducible KG-

module. Let W be the natural KG-module. If H acts irreducibly on V , then one of

the following holds:

(i) H is connected and either V = W

τ

for some τ ∈ Aut(G), or (G, H, V ) is

described by [23, Theorem 2] of Seitz;

(ii) H is a disconnected geometric subgroup of G and (G, H, V ) appears in

[6, Table 1];

(iii) H is a disconnected almost simple subgroup of G and either V is the

natural KG-module (or its dual), or (G, H, V ) is one of the cases in Table

1.