By Dr. Leslie Cohn (auth.)

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**Example text**

Suppose that b E~. (X g(~ , b E~). Then F(b) = O. 1) F(bac) = ~(a)o~F(b) (b ~ , Proof. (H e O~). c ~7)2, a ~(I). From property 2) above, it is clear that if X e ~, F(X) = 0. Property B) implies that the kernel of F is a left ideal in 9 . Hence the kernel of F contains ~ , Let ~ ( J ) = C + ~c ~ ( J )analogously. F(c) = c ~ (c ~ ) . as claimed. 1), we show first that If c = V e 9T~, we find using 2) that F(V) = - V = V ~. Assume that F(c) = c ~ if c a ~ ( J ) ; and assume that c ~ o~(J), V c Ogq.

A £ . as follows: if ~, ~ e 07 , we say that X < ~ if 141 < I~I or if Ixl = I~I and X < ~. a total order o n ~ . A l s o , properties the ordered semi-lattice mentioned above: i) Let < be the Clearly, < is (A,~) has all the if Xi, X2' ~ c A and XI < X2' then We 57 kl + ~ < X2 + ~; 2)< is a total order on A; and 3) most finitely many predecessors. natural numbers; but if dim(~ > each k e A has at (Hence, A is order isomorphic to the i, it is not isomorphic to the natural numbers as an ordered semiEroupo) The rin~ ' ~ H has the structure of a graSed rin~, indexed by A, as follows: ~ o(a)¢ = ~ @A~,~ = expu(loga)¢ , where~,w (a ~ A).

1. I f Z E ~, xT(z ~ l) ~ (~) =-ZB(Z,Vjnl(XT(I ® Vj)~)(~) - (q(Z)~)(~)° J S i0. 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.