6 I. HECK E L-FUNCTIONS
subgroup; more precisely, we introduce the group
Gs
= l[Hv,
V(£S
which by Tychonoff's theorem is compact; then we clearly have
Gs
= J]
°v
x °S-
V£S
To consider
Gs
as a subgroup of G, we identify the elements of
Gs
with those of
G whose components at the v G S are the identity element in Gv.
An immediate observation that follows from the nature of the neighborhoods
N \\v Nv defined above is that a subset C of G is relatively compact if and only
if it is contained in a neighborhood of the type B = Y[v Bv, where each Bv is a
compact subset of Gv for all v and Bv = Hv for almost all v.
Characters and Quasi-Characters
By a quasi-character of a group G we shall mean a continuous homomorphism
from G into the multiplicative group of non-zero complex numbers. The set of all
quasi-characters is denoted in the following by
#om
c
(G,C
x
) .
In particular, the dual group of G, i.e. the set of all character of G, corresponds to
the subgroup
G = #om
c
(G,T),
where as usual T is the circle group, i.e. the set of all z G C x with z z = 1.
If G is the restricted direct product of the groups {Gv} with respect to the
open compact subgroups {Hv}, then the natural injection iv : Gv c - G induces a
restriction map
Homc(G,Cx)
-
Homc(Gv,Cx)
given by x ^ Xv) where \
v
is the restriction of \ to the subgroup Gv. The dual
group G of G can be described as a restricted direct product. We prove first a
general factorization theorem for quasi-characters.
LEMMA 1. Let \ Homc(G, C x ) . Then we have:
(i) The restrictions Xv G Homc(Gv, C x ) are trivial for almost all v.
(ii)
x(a)
Yiv Xv(av), where Xv(^v) = 1 for almost all v.
Proof. Let U be a neighborhood of 1 in C x which contains no multiplicative sub-
group other than {1}, i.e. a small disk about 1 in the complex plane would suffice.
By continuity of the homomorphism x : G ^^ C x , we can find a neighborhood
N = Ylv Nv of 1 in G such that x(^0 C U. We now select a finite set S containing
all those v where Hv ^ Nv. Then Gs C N and x(Gs) C U, and hence x(Gs) = 1,
that is to say x{Hv) = 1 for all v ^ S. If a = (av) is a fixed element of G we impose
on S the additional condition that a = (av) G Gs, and write formally
aS
= (aw), with aw 1 if w G 5,
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