Get Analytic Theory of the Harish-Chandra C-Function PDF

By Dr. Leslie Cohn (auth.)

ISBN-10: 3540070176

ISBN-13: 9783540070177

ISBN-10: 3540372989

ISBN-13: 9783540372981

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1. There exists a unique linear mapping FI:T(~l) ~ ~ satisfying the following condltions" @ C[v] 43 1) FI(~I{)(1) = 1; • 2) FI(~I{)(X) = ~I~+O,aj>B(X,Hj 3) FI(X ~ b) = FI(b)FI(X ) + q(X)Fi(b ) Proof. ) + ¢I(XI~) - [B(X,Vj~)Vj (X e C~l , b e T ( ~ I (X cC~ i); )). Conditions i) and 2) define Fl(b) for b E To(~l) = ~ and b e Tl(~l) = ~ l ' respectively; condition 3) enables us to extend F I inductively to all of T ( ~ l ). Let w, Wl, and #2 be the projections of ~ onto~,7~, and respectively corresponding to the direct sum decomposition mso, define linear ~ps F(1) and ;(2):01 +97~(~ ~ ~[~] ~s follows: F(1)(~I~)(X) = [B(X,Hj n) " and F(2)(~fg)(x) = [B(X,Vj~)Vj (x ~o~ ).

I); 48 Now take 1 = ~ ~ and fix a pol~rnomlal function J E ~ satisfying the following conditions: i) J # O ; ii) there exists a linear mapping t: ~ + ~ We recall that if e~(H(~)) E ~ e such that ~ and J(~) is an irreducible factor of , then J(~) satisfies conditions i), ii), and iii). , the set of J s ~ N satisfying conditions i), ii), and iii) is a multiplicatively closed subset o f ~ . 7). Let C~(N:M:V) denote the space of functions ~:~ ÷ C~(M:V) such that the function V(~Im) is in C'(N×M:V).

B, Ad n(~)'iHj) = B(Zn'l, AC n(~)'IHj-Hj) = B(Z ,Hik(~')) - B(Z ,HiN) " - B(Z,H j ~ ). 3. li. is such that e x(H(~) ) is a polynomial function; and suppose that e ~(H(~)) = ~ji(B)bi i=l (bill) 35 is the factorization of e k'H'-''( ( ~ into irreducible polynomial functions. ,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 ~ .

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


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