EXCEPTIONAL ALGEBRAIC GROUPS

13

Proposition 1.9 Let X — A2, and let a, 6 £ { 0 , 1 , . . .,p — 1}.

(i) Ifa + b + 2p then Wx(aXi + 6A2) is irreducible.

(ii) IfWx(aXi + bX2) is reducible, then it is indecomposable with two composition

factors, Vx(aX1 + b\2) and Vx((p - b - 2)X1 + (p - a - 2)A2).

(iii) Wx(aXi + 6A2) has dimension \{a + 1)(6 + l)(a + 6 + 2).

Proof. Part (iii) is the Weyl degree formula, and (i) and (ii) are easy applications

of the sum formula in [And]. •

A similar application of [And] yields the next result.

Proposition 1.10 Let X — B2 and a, 6 £ {0,. . ., p — 1}.

(i) If2a + b + 3p, or if a = 0, then Wx(aX\ + 6A2) is irreducible.

(ii) Wx(aXi) is irreducible unless \{p — 3) a p — 3, in which case Wx(aXi)

has two composition factors, Vx(aXi) and Vx((p — a — 3)Ai).

(iii) Wx(aXi + 6A2) has dimension | ( a + 1)(6 + l)(a + b + 2)(2a + 6 + 3 ) .

Proposition 1.11 LetX,X be one of the following:

X = Ar (r 3)

X = Br (r 2)

X = Dr (r 4)

A = cAi, A2 or Xs(0 c p — 1)

X = Xi(p ^ 2) or Xr

X — Ai, Ar_i or Xr

X = E6,E7: \ = \1,\7(resp.)

Then Wx(X) is irreducible.

Proof. Except for the case where (X, A) = (A

r

,cAi) or (f?

r

,Ai), the result is im-

mediate since the Weyl group of X is transitive on the weights of Wx(X). The case

(A

r

,cAi) follows from [Sel, 1.14]; and for the other case, Wx(X\) is the natural

module for Br, which is irreducible. •

Propositio n 1.12 Let X, A be one of the following:

_X_ A

G2 Aj (p ± 2), A2, 2Ax (p # 2,7), Aj + A2 (p # 3,7)

A3 \t + A3 (p / 2), Aj + A2

B3 (p ^2) A2, 2A3, X1 + A3

C

3

( p ^ 2 ) Alt A

2

( p # 3 ) , A3, Aj + A2

C4(P^2) Ax, A2, A3

^4 A1? A3, A4, A

2

( p ^ 2 ) , Ai + A

3

( p # 2 ) ,

Ai + A4 (p 7^ 2), A3 + A4 (p ^ 2)

F4 A4 (p # 3)

T/zen

WA"(A)

has no trivial composition factors.