supported smooth functions on the group G(A). This feature of our formula has a
novelty for our purpose, because the limiting measure λS
in Theorem 1.1 is almost
evident from the term Ju(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
when regarded as a distribution on G(Fv), satisfies the differential equation (Green’s
Ψvz)(s) (
λv(s)Ψvz)(s),f (
H Kv
dk, f Cc
with λv(s) = (1
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) (
with λv(s) = qv
+ qv
, where Tv is the Hecke operator corresponding to
the double coset Kvdiag( v, 1)Kv, and, for an ov-ideal a,
Φaz) (
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
is defined to be the product of the Green functions
Ψvz)(sv; (
gv) over the places
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