Setting g = 0 we obtain the Poisson summation formula:
(PSF) $o = £*(')•
This formula will be valid for $ of Poisson type with respect to the pair (G, L) and
with a Haar measure d\i such that vol(G/L) = 1.
We now make a remark that will play an important role in the number theoretic
situation. Suppose $ is a function of Poisson type with respect to the pair (G, L)
for which at least one side of (PSF) is not 0. Call dp the dual measure to d/i; put
dp = c, and perform the calculation leading to (PSF) with the roles of G
and G interchanged, starting with the Fourier transform of $ with respect to the
Haar measure
on G; by the inversion formula this is c
- 1
$ , and we get the
same Poisson formula with $ replaced by c_1I. By comparing the two formulas we
get c = 1. It follows that the measures d\i and dp on G and G, normalized so that
vol^(G/L) voljx{G/L^) = 1 are dual to each other. In the number theoretic case
of interest, where G is identified with G by means of an isomorphism which sends
L onto
the self-dual measure d\i on G will be that for which voll_i(G/L) 1.
Our basic examples of functions of Poisson type will come from spaces of
Schwartz-Bruhat functions.
Measures on Restricted Direct Products
Let G be the restricted direct product of the locally compact groups {Gv}ves
with respect to the subgroups {Hv}vesi where the Hv are compact for almost all
v. For each v let dfiv be a measure on Gv such that JH d\xv 1 for almost all v.
We now show how to associate with this data a unique measure on G. Since by
definition G is the inductive limit of open subgroups of the form
Gs=l[Gvx Y[HV,
vES v£S
where 5 is a finite subset of S, it suffices to construct a family of coherent measures
djis on these subgroups. The choice of measure on Gs is
is that measure on the factor
= Ylvds Hv for which
/ dfis = TT / d/iv (a finite product!).
To verify that the dfis are coherent we must show that for any two finite sets T and
S with T D S, in which case GT 3 Gs, the restriction of dfir to Gs coincides with
df^s- It is clearly sufficient to verify this for two subsets T and S with T SU{w}.
Now, since
Gr = ±lw X Gr ,
we have that
d\i x
is the required measure on
We have therefore
dfis = TT dfiv x dfiS
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