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.
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