Analytic Theory of the Harish-Chandra C-Function by Dr. Leslie Cohn (auth.)

By Dr. Leslie Cohn (auth.)

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Preservation of Certain Filtrations Suppose that R is a ring, and that A is an additive will say that R is a A-graded ring each I e A, we have an additive l) RIR ~ ~ RI+ ~ 2) E = ~ ® leA (1, ]j s semi~roup. (or simply a graded ring) if for subgroup R I of R such that A) and . R 1 Suppose again that R is a rin£ and that A is an additive Suppose also that A is partially ordered by a relation lI < 12, then 11 + ~l < 12 + ~ (I 1 , 12, p A-filtered ring if we have additive i) RIR~ 2) R I -~ R p if I < ~, and 3) U R I leA R I+~ (I, ~ e s A).

Element such that r e R k but r ~ R~ if If M is a filtered R-module, we define the notion of the leading term of elements of M analogously. Now let A ~ G ~ * be the semi-lattice If I = ~mi~ i c ~ i , generated by Eo(P,A). we define the level Ixl of X to be [m i. usual l e x i c o g r a p h i c o r d e r on ~ now define a new order < on O l d e t e r m i n e d by t h e r o o t s ~ l , . . , 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 ~ .

Let w, Wl, and #2 be the projections of ~ onto~,7~, and respectively corresponding to the direct sum decomposition mso, define linear ~ps F(1) and ;(2):01 +97~(~ ~ ~[~] ~s follows: F(1)(~I~)(X) = [B(X,Hj n) " and F(2)(~fg)(x) = [B(X,Vj~)Vj (x ~o~ ). 2. If X1, X 2 E ~ (X ~O~l). , then q(xl)/l)(x2) - ~(x2)/1)(xI) = F(1)([Xl,X2]). Proof. , ~-B'HJ ]~) : - ~B([~l(X2n-1), w2(Xl~'Z)] , Hi). 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]).

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