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A dts"IA ... 3 apply, in particular, when c ~(M,TM) (the space of TM-Spherical Schwartz functions on M) or when ~ is constant (since the constant functions on M belong to C~(M,~M ) where ~ is the trivial one-dlmensional double representation of K). For in both cases ~ is bounded on M. § 7. The ~epresentatio~ q Definitions. 1) let q(X) denote the operator on C'(~) given by (q(x)f)(~) = . ~ cp+ B(X,~8)f(n;X 8) (f e C®(N), ~ e N). 2) (f e C®(N), ~ e ~, m e MA). i. i) q is a representation of ~ ring C=(~).

Hence, ~(Xl);(z)(~i~)(x2) - q(x2)F(z)(~[~)(xz) = ~B([~2(Xl['Z) , ~z(X2n-l)] - [~2(x2n-l), ~l(Xln'l)], Hj) = [B([Xl~'z,x2~'z],Hi) = F(Z)(~I~)([Xz,X2]). 3. If XI, X2 e~ , then q(XI)F(2)(X2) - q(X2)F(2)(XI) + [F(2)(XI), F(2)(X2)] = F(2)([XI,X2]). xz' ×2 ~ ' q(xz)F(2)(~I~)(x2)= - X B(×I,XB~)~(X2'[x-B'vj]~)vJ = - [B(X2~'I , [~2(XI~-I),Vj])Vj = ~([~2(XZ~-I),X2~'Z]) = w([~2(Xl['l),~I(X2['I)]). Hence, 45 q(XI)F(2)(~I~)(X2) - ~(X2)F(2)(~I~)(XI) + [F(2)(~I~)(XI) , F(2)(vI~)(X2 )] = ~([~2(XI{-I), ~l(Xl{'l)] .

Proof. Immediate. ~12. 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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