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

By Dr. Leslie Cohn (auth.)

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Clearly, each G(~)I is a /[M-mOdule; and G(~) is generated as a ring by {ZsI~ z~inl. P÷} and G(~) o = ~ M " ns Also it is clear that the "standard monomials" (ni>_O) form a basis for G(~) as a right ~IM-mOdule. ~B ns01ml'''0p mp S (ni >. > O) form a basis for G(~) as a C-vector space. p). ,p,~ ~ P+). x' ~'<(YI÷Y2)IO~ Hence, [ZTI,Z¥2 ] = I/2B([XTI,Xy2],X_yI_Y2)ZYI+y 2 (Tl,y2 e P+). This • 60 ) shows first that~Cis a Lie subalgebra of G~p~), and second that the linear map ¢ : ] ~ e ~ M ¢(X B) = 2Z B (seml-dlrect product) ~ G(~') such that (B E P+) and c(Ui) = Ui isomorphism of ~ "C e ~M onto~.

Therefore Y - 2H 0 = O, as claimed. 2. Suppose that Z s ~ function on ~ of re dmced degree d. , £) and 40 eC (M,TM) , then x(z)fNF(~)eiV'o(H(~))~(~lm)d[ = [j~lfNB(Z, Hj~)F(~)eiV-o(H(~))~(Elm)d~ - fN{[sEp+B(Z, XB[)F(~;X B)}eiV'0(H([))~([lm)d[ - Zj;~B(Z, V~F([)eiV-p(H(~))(XT(I @ Vj)~)(~Im)d[, all integrals being convergent. ,£). 2, all of the above integrals are convergent. Let V(R) be the subset of ~ defined in §6; and let fs(v]~IZlm) B(Z, X~)F(~)eiV'P(g(~))0(~ m). I). Now let R ~ ~. 3, the last integral converges uniformly to O; and so the proposition follows.

8s I= 0 Clearly, ve may n assume that F(~) ~ ~yep+tT(~) 7, where ~Tcp+7(Ho)n 7 < d + B i(Ho) = d i. Set B~(d) = ma~ l J=l,,.. zm~v,~j> >_B~(d). (~)l"s. Adt6 I. A dt 8s ] = C'Rs-l. Adtsi i I(R) denote this integral. 5). Hence, Vsi(R ) = {[ ¢ ~Itsi(~ ) = +_2, Its(i)[ ~ R if (5# 8i }. xR . xRxR = RldZlXvol(A R s. xl R ±) s ×R 1 = e o n s . xR s'l. Hence, in either case, A IF(~) lexpT-(Imv÷0) (H(~)) } IdtsiA ... A dts"IA ... 3 apply, in particular, when c ~(M,TM) (the space of TM-Spherical Schwartz functions on M) or when ~ is constant (since the constant functions on M belong to C~(M,~M ) where ~ is the trivial one-dlmensional double representation of K).

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