188 Arthur, Fourier transforms of weighted orbital integrals

in terms of the distributions (2.1), (2.4) constructed in §2 by means of normalized ators. The normalizations implicit in the two terms Jt^ix^.fq) and Jt^i^^gq^) are stood to be complementary.

To prove the claim, we recall from [8], §4 that

Л« ® 7C2,g X/) = itiMnl ® 7Г2, P)M^Ï ® ^2^8 X/)) ,

if n^ and 7C2 are irreducible constituents of the induced representations

( T^ = ^W, Äe^^(Li), of L(jF). Here, A(^i ® tcjjP) is the operator

lim X /QiA,nl®n2,P)eQ{Ar'

obtained from the (G, L)-family

/ ^ ( Л , ^ ; ^ (8)712,P) = (/о|р(я;^)(8)/е,р(7С2))"Ч^о1р(^1.-л)®^|р(7^2,л)).

A € ial, Q G ^(L), Let {^q(A, n^ ® 7Г2, P)} be a second (G,L)-family obtained by ing complementary normaUzations to the intertwining operators. That is,

fQiA , nl ( S>n2 , P ) = rQ(A,nl®n2,P)^Q(A,n^,®n2,P), where

Гд { А , п1 ® Я2,Р) = (го|Р«)^о|р(71:2))"Ч'о|р(^1,-л)'о|р(^2.л)) • Now

by the properties (r. 5) and (r. 1) of the normalizing factors. Since n^ and П2 are irreducible constituents of (x^, the corresponding normalizing factors are the same as the ones for a^. Therefore

'ölp ( ^i ) ^Q|p ( ^2 ) = 'p|ö(^^)'q|p(^^) = 'pipC^^) • It follows that

Гд ( А , п1®П2 , Р ) = Гр,р((т^)"^Гр,р(о:[),

a function which is independent of Q, and which equals 1 at Л = 0. We conclude that Л«®7Г2,^') equals

^L«®^2 . ^ ) =lim X ^ö(^'^i®^2.^)Öq(^)"'-

In Other words, the weighted character J^inl ® 7C2,g x/) above may also be constructed from normaUzed intertwining operators, provided that we use complementary normaUza-