6 1. INTRODUCTION

supported smooth functions on the group G(A). This feature of our formula has a

novelty for our purpose, because the limiting measure λS

η

in Theorem 1.1 is almost

evident from the term Ju(n|α)

η

+

Jη(n|α)

¯ u

occurring in the geometric side given in

Lemma 12.1. We should remark that, contrary to [10], our framework includes the

case when η is trivial.

Here is a brief explanation of the structure of this article.

In §2, after introducing notation for fundamental objects like Haar measures,

characters and representations for various local or adelic groups, we study period

integrals of automorphic forms on G(A) in connection with the L-functions of au-

tomorphic representations. In the final part of §2, we review a basic theory of

Eisenstein series including a computation of intertwining operators for several spe-

cial vectors of the principal series.

In §3, we prepare lemmas necessary to discuss the convergence of series and

integrals for functions on G(A) with left H(F )-invariance. The subsection 3.3 is

devoted to the study of a space consisting of moderate growth functions on the dis-

crete quotient of G(A); in particular, we establish a density of compactly supported

functions (Lemma 3.8), which plays a crucial role in the proof of Lemma 8.2.

In §4 and §5, we introduce Green’s functions on G = GL(2) over a local field.

When the field Fv is archimedean, such functions were already studied in [54].

Recall that they form a family of continuous functions

Ψvz)(s; (

gv) depending on two

complex parameters (z, s) and satisfying the equivariance condition

Ψvz)(s; (

hgk) =

χz(h)Ψvz)(g) (

for h ∈ H(Fv) and k ∈ Kv with χz being the quasicharacter of H(Fv)

defined by χz(diag(t1,t2)) = |t1/t2|z, and most importantly, each function

Ψvz)(s),(

when regarded as a distribution on G(Fv), satisfies the differential equation (Green’s

equation)

Ψvz)(s) (

∗ Ωv −

λv(s)Ψvz)(s),f (

=

H Kv

f(hk)χz(h)−1dh

dk, f ∈ Cc

∞(G(Fv))

with λv(s) = (1 −

s2)/4,

where Ωv is the Casimir element of G(Fv) (see §4). This

kind of function on the upper half plane have been used classically to construct the

resolvent of the hyperbolic Laplacian for automorphic forms ([18]), and its higher

dimensional analogue for non Riemannian symmetric spaces was studied by [39]

and [40] to obtain the automorphic Green current for arithmetic cycles on locally

symmetric spaces of unitary groups. Over a non archimedean field Fv, we introduce

a similar function

Ψvz)(s; (

gv) in analogy with the archimedean counterpart as a

function on G(Fv) with the same (H(Fv), Kv)-equivariance as above such that it

satisfies the inhomogeneous equation

Ψvz)(s) (

∗ Tv − λv(s)

Ψvz)(s) (

=

Φoz),v,v(

with λv(s) = qv

(1+s)/2

+ qv

(1−s)/2

, where Tv is the Hecke operator corresponding to

the double coset Kvdiag( v, 1)Kv, and, for an ov-ideal a,

Φaz) (

,v

denotes the unique

function on G(Fv) supported on H(Fv)K0(a)v and satisfying

Φaz) (

,v(hk) = χz(h) for

any h ∈ H(Fv) and k ∈ K0(a)v (Lemma 5.2).

In §6, we introduce various kernel functions on G(A) such as the adelic Green

function and its smoothing. Given a finite set of places S containing Σ∞ and a

family of complex numbers (z, s = {sv}v∈S ), the adelic Green function

Ψ(z)(n|s;

g)

is defined to be the product of the Green functions

Ψvz)(sv; (

gv) over the places