10 2. PRELIMINARIES
Haar measure
d×xv
on Fv
×
by
d×xv
= cv dxv/|xv|v, where cv = 1 if v Σ∞ and
cv = (1 qv
−1)−1
if v Σfin. We fix a Haar measure of the idele group

by
d×x
=
v
d×xv.
For y 0, let y

be the idele such that y
ι
=
y1/dF
for
all ι Σ∞ and y
v
= 1 for all v Σfin. Then, y y is a section of the idele
norm | |A :

R+,

which allows us to identify

with the direct product of
R+
×
= {y| y 0 } and the norm one subgroup
A1
= {x
A×||x|A
= 1 }. We fix a
Haar measure
d1u
on
A1
by requiring

f(x)
d×x
=
+∞
0 A1
f(y u)
d×y d1u,
f
L1(A×).
We fix Haar measures dhv, dnv, dkv and dgv on groups Hv, Nv, Kv and Gv
respectively by requiring vol(Kv;dkv) = 1 and
dhv =
d×t1,v d×t2,v
if hv =
t1,v 0
0 t2,v
, t1,v, t2,v Fv
×
,
dnv = dxv, if nv = 1 xv
0 1
, xv Fv,
dgv = dhv dnv dkv if gv = hv nv kv (hv Hv, nv Nv, kv Kv).
The adele group GA is decomposed to a direct product of GA
1
= {g GA|| det g|A
= 1 } and A = {y 12| y 0 }. By taking the tensor product of measures dgv on
Gv, we fix a Haar measure dg on GA; then the decomposition GA = GA 1 A yields
the Haar measure d1g on GA 1 such that dg = d1g d×y. By virtue of the Iwasawa
decomposition,
G1
A
ϕ(g)
d1g
=
+∞
0 x∈A
(u1,u2)∈(A1)2
k∈K
ϕ
1 x
0 1
tu1 0
0
t−1u2
k
t−2 d×t d1u1d1u2
dx dk,
(2.1)
where dk is the Haar measure on K with total mass 1.
2.3.1. Let g∞ be the complexification of the Lie algebra of G∞. The universal
enveloping algebra of the complexification of a real Lie algebra l is denoted by U(l)
and the center of U(l) is denoted by Z(l). Let g∞ and denote the Lie algebras of
G∞ and Gι, respectively. Then, U(g∞)

=
ι∈Σ∞
U(gι) and Z(g∞)

=
ι∈Σ∞
Z(gι).
For any function ϕ on GA, the right translate of ϕ by an element g GA is denoted
by R(g)ϕ, i.e., [R(g)ϕ](x) = ϕ(xg). For a smooth function ϕ on GA, its right-
derivation by a differential operator D U(g∞) is also denoted by R(D)ϕ.
2.3.2. Let v Σfin. The Hecke algebra for the pair (Gv, Kv) is denoted by
H(Gv; Kv). Recall that it consists of all the C-valued compactly supported func-
tions on Gv constant on each Kv-double coset; it has the unit given by 1Kv =
vol(Kv;dgv)−1
χKv . We define an element Tv H(Gv, Kv) by
Tv =
1
vol(Kv;dgv)
χKv[
v
1
]Kv
.
2.4. L-functions for idele-class characters
We refer to [49], [42].
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