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