16 DAVID C. VELLA

W

W T

ei(X) J,a. But ^(X) i s in (Aj)+ so ^ ^ X ) ,ct * 0, and t h i s shows the low

W

J J

weight X of Sj(X) i s a negative of a weight of A+. Thus we could jus t a s

well parametrize the irreducible Pj-modules by thei r low weights. For

X € A+ there i s a unique irreducible Pj-module with low weight -X and we

denote thi s by Mj(-X). The high weight of Mj(-X) i s then (-X) J . The map

W

J

£:A -» A given by £(X) = (-\) i s a relativ e versio n of the opposition

involution. If Sj(X) has high weight X then i t s dual Sj(X) has low weight -X

s o Sj(X) = Mj(-X) = Sj(e(X)). Thus, when J = A, e(X) = (-X) u = X i s the

usual opposition involution and S(X) = S(X ), while if J = 0, then £(X) = -X

reflectin g the fact that the dual of the one dimensional B-module X i s the

module -X.

Another way to view £ i s a s being obtained from the opposition involution

on Aj. Write X = e2(X) + e2(X) and then note e(X) = -e

1

(X)

J

- e2(X) where

-E1(X)

J

= e2(X)* in the lattic e Aj.