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

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By Dr. Leslie Cohn (auth.)

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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)] . ~I(XI[-I)] + [~(XI~'I), X2~-I)] ) = ~([XI~-I,x2~-I]) = F(2)(vI~)([XI,X2]), as claimed. b. If XI, X2 e ~ l , then q(Xl)¢i(x2) - Q(x2)~i(xi) = ~/[Xl,X2]).

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~~ , ~ .

1). Consequently, Y [P+ IF(~)lexp{-(Imv+O)H(~) } < C, exp {-20H(~)}, which is Integrable on 9. Let V(R) be the set of elements ~ E ~ such that I t (~)I < R for all y E P÷ (R > 0). 3. Suppose that F(~) is a polynomial function on ~ of reduced degree d. Suppose that ~ E C~(M,TM ) and that ~ is bounded, and let f(~l~Im) = F(~)ei~-0(H(n))~(~m) (~ s C~e, ~ ¢ S, m c M). ,~. ~Proof. 1, Let IR(~:m) = ]V(R)f(~l~;X_yIm)d~. AdtBs 1 (where VB(R) = {~ e HIIts(~) I = R, It6(~) I ~ R for 6 e P+, ~ ~ 8 or - @S}).

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