By Dr. Leslie Cohn (auth.)

ISBN-10: 3540070176

ISBN-13: 9783540070177

ISBN-10: 3540372989

ISBN-13: 9783540372981

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**Additional info for Analytic Theory of the Harish-Chandra C-Function**

**Example text**

A). ;a and Z ~ ~ , Remark. (Z e ~ , ~ £ ~). Ji divides q(Z)J i. Let Ji(~) and Cj. be as in the preceeding corollary. (VI~) = 0 for V E ~ M ' ~ e N. (~) = I when ~ = e. ,a) such that p(m)Ji(~) = Xi(m)Js(m)(i)(~) for all ~ e ~ and m e ~ . 0(m)Ji(~) = Js(m)i(~). m + ~m)Ji({) Letting ~ = e, we see that ~(m) m i; hence, It then follows that the map ~ + ~ sending (5 E H fixed) assumes only finitely many values, hence, 36 being continuous, is identically equal to Ji(~) on ~ , then p(V)J i = O for V e ~ .

J(Zy) e i ~ Therefore, Fj(~)(Zy) ¢ )/ @ ~ with (k = YI~). Z ). 1 V2 ~J with X = (y2+'''+Yj)IC~; and with ~ = yll~. Hence, since the submodules (X e A) define a filtered ring structure on @ 7 @ , @ . • clear that Fj(v)(Zy~.. Zyj)Fj(~)(Zy1) ¢ / ) : @ ~ ,,. +~j)16~.

Let d(b) denote the reduced degree of Fj(~I~)(b) as a polynomial function on N; and let Cj(Zb) = B''(c(b) + d(J)). ,£), T(Zb) fN-J(~)eiv-0(H(~) )~(~m)dn £ = ~j=l~i~+P,~>fNJ(~)ev(n)B(Z, Hj~)(~T(I @ Fj(~l~)(b))$)(~Im)d~ + f~(q(Z)J)(~)ev(~)(~x(l @ Fj(~l~)(b))~)(~Im)d~ + /~J(~)ev(~)(~ (i @ q(Z)Fj(vl~)(b))~(~Im)d~ - ~N-J(~)ev(~)~B(Z, vjn)(IT(l ® Fj(~l~)(b)Vj)$)(~Im)d~ = ~J(~)ev(~)(Ix(l ® {Fj(~l~)(b)Fj(vl~)(Z) + q(Z)Fj(vl~)(b)})~)(~Im)d~ = /~j(~)ei~-P(H(~))(~ (i ® Fj(vl~)(Zb))~)(~Im)dH. 50 Hence by induction the proposition is valid for all b g ~ .

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

by Paul

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