By Dr. Leslie Cohn (auth.)

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M is defined by the subspaces ~ " X~M' where ~V)'~x ={b c ~ Iba = expk(Ioga)b (a c A~ (X ~ A). Clearly, the monomials X B1nl .. 2. The submodules ~k @~M~ ~ a filtration of ~ (I c A) of ~ @ o ~ @ ~ M ~ by flnltely-generated free ~L-modules. associated graded module is isomorphic as right ~/~u-module to ~ Proof. ~. 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

3. The mapping Fj: [,~ 8~M~'/2--~ ~'~ ~ ~ 8 C[,j] is a homomorphlsm of filtered~i~-modules. Proof. +yj)]~ @ with ~J X. We proceed by Induction on J. If J = i, b = Zy with y c P+. 6, Fj(~]~)(Zy) - Cj(Zy]~) is an~-linear combination of polynomial functions of the form B(Zy,V ~) with V e ~ G ~ C ~ . /2B(Xy,~) and 62 B(X ,V ) = B(X Y Fj(v)(Zy) - Cj(Zy)~ = expy( '~@bt~ so clearly with X ~ We claim that @j(Zy) ~ ~ • N with X = YI~. For let J(~) = ~eAJ (~) and Sj(ZI~) = ~weACj, (ZIH) be the decomposition of J(~) and Sj(ZI~) (Z c ~ decomposition ~ ) into their homogeneous components according to the = ~ @ b~~ , ~ .

I). Now let R ~ ~. 3, the last integral converges uniformly to O; and so the proposition follows. ~ii. The Functions Fj(vl~)(b) Let ~ l be a subalgebra of ~ with universal enveloping algebra ~l; and fix a pol~vnomlal function I ~ ~ i) ii) ~ satisfying the following conditions: I#O; there exists a linear mapping ¢I:~i ~ such that functions on ~ and extend the representation q of ~ on 6 ~ (57) to ~ ~ ~ ~ b~ sett in~ q(x)(c¢) -- cq(x)¢ (z ~o~ , c ~)9l, ¢ ~ ( ~ ) . ,£). 1. There exists a unique linear mapping FI:T(~l) ~ ~ satisfying the following condltions" @ C[v] 43 1) FI(~I{)(1) = 1; • 2) FI(~I{)(X) = ~I~+O,aj>B(X,Hj 3) FI(X ~ b) = FI(b)FI(X ) + q(X)Fi(b ) Proof.

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