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16). i) of [I] p = -d + z /~(~'l~2,~'c+z)~(~'l~2,~'d+Z)Cl(-C-d+2z,~) w I (~z'c)w 2 (-1)~ (w-l,-z) = If C - z ~ 0 o=~ 2 • t h e n the o n l y n o n z e r o So w e t e r m o n the l e f t - h a n d s i d e is for find that ~ ( ~ 2 ~ l)' ~ ' d + Z )zC l ( - C - d + 2 z ' ~ 2 This implies ~ 0 o that - c - d + 2 z = -c t h a t is d ffi 2z . that either v = z or v = 0 z = I ~ c > 2 are not compatible. Similarly, 2 and ~2 if Therefore d - z < 0 We choose and and z = I . 18). the r o l e s o f implies that z > 0 .

Here fl is given explicitly by fl(a) = q'V~l(w)(pl(a) if a E ~"VR = 0 otherwise. 2) is satisfied. 14: We assume that nI = ~ l , V l ) and ~2 = ~(~2'V2 ) with n = O~ I ) = O~ 2) > 0 = 0~11~2) = O(m) , 0(~;I ) = 0(92) = 0 . z corresponding to ~oi (y) " defined by the following conditions: ~0l(ae) = ~0l(a)~ I(¢) for ¢ in RX , [ ~iCw)~1(a)lalS-½dXa= LCS,~l)e(l-s,~I) ; ~02(a¢ ) = q02(a) for ¢ in R× , ~°2(a) lalS-~dXa = LCs,v2)¢Cl-s,~2 I) , In ~(s,WI~W2,~) we integrate for t in Rx and x in R X ° We find in that way ~0(-I)~ (s ,WI,W 2,~) = ~ ~I (w)~°l (a)~2 (w) f2 ('a)dXa where f2 is the element of f2(a ) = ~R x ~2~I ~(~2,~) defined by x~,,)V, "~ 2-I (x)dx ~P2(a) .

V21)-1 . u)~2 <-i)~ cS,~l~~),, <1-s,~~i t in RXm "u and x in ;i>-i. Rx . ~2~ 1) . Then we get . (-1)~7 (1-s,w 1,w 2,~) = c ( s , . ]')rt"1 (w)qO1 (a). 2 (x)dx . 2) is clearly satisfied. 15). 15: ~(~l,Vl) We assume that ~I is absolutely cus~idal or ~i = ~ and all the p~oducts ~iV2 , ~I~2 , Vl~ 2 , VlV 2 , are ramified. For ~0i in ~(~i,~) (or characters of j n R x) in Z and all quasi-characters X of we set ~i(n,x) = ~RX ~i(~n)x(e)d~ where de is the normalized Haar measure of transform of ~0i The operator ~i(w) RE .

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