By Dr. Leslie Cohn (auth.)
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Additional resources for Analytic Theory of the Harish-Chandra C-Function
B'HJ ]~) : - ~
Z~ nsH E ~ . -i Finally, Z^ Z Z^ 1 nse ~I ~i 71 ~i "''Z~s by induction on Yl" 59 Now suppose that ~ e Z(P,A). Clearly,~ M normalizes ~ 8 ~ $ ~ ; so by the preceeding paragraph, it is obvious thatJ~< ~ is invariant under left multiplication by elements of ~M" above) t h a t ~ l ~ u c ~ I+~ But then it is clear (again by the (~,u a A), as claimed. For the second statement of the proposition, recall that the associated graded ring of ~ is by definition the ring G(~) = ~k~AG(~)k, where G(~)I = ~ X / O ~ I ' .
2) It clearly suffices to show that p(m)t 8 e ~ (m e MA, 8 c P+). But if m e 5~ and ~ £ N, p(m)tB(~) = B(XB,log m'l~m) = B(~,log ~) = ZB(~,X_y)B(Xy,log ~). h) P(m)t B = EB(~B,X_y)t Y Clearly, then, p(m)t 8 ~ ~ N , 3) Suppose that V ~ 93~ @ b ~ (m c MA, 8 c P+). as required. and f e C=(N). Then (q(V)f)(~) = - f(@(V[~)) = d f(¢(exp(-tV)~))It=O = d f(¢(exp(-tV)~exp tV))It=O St St = d (p(exp tV)f)(~)It=O = (o(V)f)(~), as claimed. St h) If Y ~ and f z C=(~), (q(Y)f)(~) = - f(¢(Y~)) = - f(Y%~). 5) If X c C~d , m E MA, and f c C=(N), we have (q(xm)f)(~) = - f(¢(xmw)) = .