12

I. HECKE L-FUNCTIONS

THEOREM 10. If X is a finite dimensional vector space over R and if T is a

distribution in X with T(I) 0 for all non-negative I G C%°(X), then T = TM is

a positive measure.

If X is a finite dimensional vector space over a local field k with discrete valu-

ation, then S(X) is the space of locally constant functions with compact support.

The tempered distributions are in bijective correspondence with the simply additive

measures on the family of open compact subsets.

The Poisson Summation Formula

Let G be a locally compact abelian group. Let L be a discrete subgroup of G

so that G/L is compact. The orthogonal complement of the subgroup L is

L±

= {XeG: X(g) = 1 for all g e L}.

On G we will normalize the Haar measure d/x, so that JG/L d\i — \.

DEFINITION 17. A complex valued function & on G will be called of Poisson

type with respect to the pair (G, L) if the two series

are absolutely convergent, uniformly on compact subsets with respect to the argu-

ments g and g.

If I is of Poisson type for the pair (G, L), the function

F{g) = YJ*{9 + l)

leL

is well defined on G/L and represents a continuous function. The Fourier transform

F as a function on L1- is

F{9)= I (52*(g + l)){g,g)dn(g),

JG/L

l€L

where, as usual, g is the image of g in G/L under the canonical projection of G

onto G/L; since (g + 1,1} = (g,I), we obtain

F(J}= [ (J2^(9 + l))(9 + lJ)M9)

JG/L

leL

= [ *(g){g,i)dn(g)

JG

= Hi)-

Under the assumption that $ is of Poisson type for the pair (G, L), it follows that

l is integrable on L^ so that by Fourier inversion applied to G/L and

L1-

F(g) = ^2$(g + l}= £|.(05,0.

'€L leL-i-