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

By Dr. Leslie Cohn (auth.)

ISBN-10: 3540070176

ISBN-13: 9783540070177

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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]). 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)] .

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)) = .

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