Download e-book for iPad: Analytic Theory of the Harish-Chandra C-Function by Dr. Leslie Cohn (auth.)

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

Sample text

2Hp. Denote by Y the elem,mt [Bsp+[XB,X_B]. 8]] = ~8,yEp+B([V,XB], X_y)[Xy,X_~] + B([V,X_~], Xy)[XB, X_y] = [B,y~+{B([V,Xs], X_y) + ~([V, X_y], XB)}iXY, X_B]. By the invariance of the Killin~ form, this is zero. the center of ] 0 q e O ~ . Hence Y belongs to Since [XB,X_B] = HB, B(Y,H) = ~8Ep+B(HB, H) I = 2p(H) = 2B(H0, H) (H s O~); therefore, Y - 2H P s dT(_~ (OL) =)7b. In fact, Y - 2H in ~M" Thus, since the map e (Y) = - Y. lies in the center of 7)q, hence ([21], Vol. l, p. 72, 73) P e(Y-2Hp) = Y - 2Hp.

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~.

G,~ ~ c Consider the following direct sum decompositions o f ~ ~). : i) 2) give rise to corresponding decompositions o f ~ : These If b e , let R x (b) denote the projection of b onto ~k(x) according to the decomposition ~; and let Sx(b) = Rx(b)k(x)-l. the projection of bk(x)-I onto~corresponding If f e C®(A), let f(x) = f(a(x)). Then Sx(b) is to B)~ Then f(Z~x) = 0 for Z ¢ ~. [email protected]~.. Hence, if b e ~ ®~ (~ ® T q ~ )k(x) then f(b~x) = O. Therefore, if b ¢ ~ , f(b{x) = f(Rx(b){x) = f(Rx(b)k(x)~la(x)) = f(Sx(b)[a(x)) = f(Sx(b){a(x)); so ~(bEx) = Sx--7~(x).

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


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