16 I. HECK E L-FUNCTIONS
product is the product of the Fourier transforms of the components. Recall that if
G is the restricted direct product of the groups {Gv}ves relative to the subgroups
{Hv}ves, then the dual G of G is the restricted direct product of the groups {Gv}
relative to the subgroups {H^-}. As a locally compact group, G has a Haar measure;
we now describe how that measure is determined once a choice is made for each v
of
(i) a bilinear pairing
Gv x G , ^ T , (a,x) =x(a) ,
(ii) a normalized measure dfiv on Gv, and
(iii) the corresponding dual measure dftv on Gv.
We write a generic element of G as \ Yiv Xv Let $v be the characteristic
function of Hv:
$v(av) = charHv(av), av G Gv.
It is obvious that the Fourier transform of $v (a) is
$v (Xv) = charH± (xv), Xv Gv;
applying the local inversion formula to &v and observing that frv(av) = ^v(a~1)
we obtain the simple relation
charHv(av) = ( / dfiv)( / djlv)charHv(av)
from which we obtain
/ dfiv / dfjLv = 1.
JHV JH±
Since the first factor above is 1 for almost all v, It follows that the second factor
is also 1 for almost all v. Hence in accordance with the definition of a product
measure on a restricted direct product we may choose as Haar measure on G the
product measure associated with the choices {dfiv} and {djlv} and we write
dXv =Y[dxv
V
For each v G S, let &v be a complex valued function in L\(GV) with &v G
Li(Gv) and suppose that $v = charnv for almost all v. The Fourier transform of
the product function
V
is calculated by applying the Lemmas 3 and 4 to the function
$(a)x(a)
- 1
= YlSv(av)xv(av)-\
V
for x YlXv a character in G. I(a) is clearly continuous. In fact the integral
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