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1. Application of the Differential Equations Suppose that (P,A) is a parabolic pair of G and let p 1 denote, as usual, ~ ~aeZ(p,A)m. Then ~S~p+[Xs,X_B] Proof. = 2Hp. Denote by Y the elem,mt [Bsp+[XB,X_B]. 8]] = ~8,yEp+B([V,XB], X_y)[Xy,X_~] + B([V,X_~], Xy)[XB, X_y] = [B,y~+{B([V,Xs], X_y) + ~([V, X_y], XB)}iXY, X_B]. By the invariance of the Killin~ form, this is zero. the center of ] 0 q e O ~ . Hence Y belongs to Since [XB,X_B] = HB, B(Y,H) = ~8Ep+B(HB, H) I = 2p(H) = 2B(H0, H) (H s O~); therefore, Y - 2H P s dT(_~ (OL) =)7b.

The sequence ~'~ is an exact" sequence of finitely-generated ire# ~M-mOdules, Hence the 61 sequence Jv x is also exact. ~,]~M @~M:O~, ~ is 6: ~ @ ~ ÷ ~M " ][~ e o(~)~ ~ ~ - - " isomorphic to @°~ 1'~8~ @ 5hLM~ ~1

1. Proof: and define a right by setting Fj is a homomorphism of right'M-mOdules. We show that Fj(bd) = d~Fj(b) AS b e f o r e , we let q9 (j),, (b ¢ ~ , = --(~+ ~e + . . +%J F i r s t we note t h a t F j ( ~ ] H ) ( V ) = -V d EC~M). (J_>O) and let (V ~ ~M). This follows from the f a c t t h a t 1)B(V,Hj H) = B(V,Hj) = O; 2) )j(vlH) = 0; and 3) -TjB(V,VjE)Vj = -•jB(V,Vj)Vj = -v. We claim that Fj(d) = d % holds if d E~(l! d C~)M (d e ~ M )• We have Just shown that this Assume it to be true if d • ~M(n); (n), V ~ ~M .

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