2 1. INTRODUCTION
The set S(n) is disjoint from S(f) Sfin and has the even cardinality.
ηv( v) = −1 for any v S(n).
For any ideal n satisfying these conditions, which are sufficient for the sign of the
relevant global -factor to be +1 (Lemma 2.3), let Πcus(n) be the set of all the
irreducible cuspidal representations π of GL(2, A) with trivial central characters
having non zero K∞K0(n)-fixed vectors, where K∞ is the standard maximal com-
pact subgroup of GL(2,F ⊗Q R) and
K0(n) =
v∈Σfin
{
av bv
cv dv
GL(2, ov)| cv nov }.
Let π Πcus(n) with n IS,η.

Then, there exists a family of irreducible smooth
representations πv of GL(2,Fv) for all places v such that πv is unramified for almost
all v Σfin and such that π

=
v
πv. When v Σ∞ or v Σfin prime to n,
then πv
is isomorphic to an unramified principal series representation
Iv(||vv/2)ν
(see 2.5.1) with the parameter νv belonging to the space Xv 0+, where Xv 0+ denotes
iR+ ∪(0, 1) or i[0, 2π(log qv)−1]∪({0, 2πi(log qv)−1}+(0, 1)) according to v Σ∞ or
v Σfin, respectively. Note that the set of Kv-spherical unitary dual of PGL(2,Fv)
coincides with the union of
{Iv(||v)ν/2|
ν Xv
0+
} and the set of unitary characters
of PGL(2,Fv) trivial on Kv, where Kv is the standard maximal compact subgroup
of PGL(2,Fv). The spectral parameters at S of π is defined to be the point νπ,S =
(νv)v∈S lying in the product space XS
0+
=
v∈S
Xv 0+. We endow the set XS
0+
with
the induced topology from CS. On the one hand, the set Πcus(n) defines a Radon
measure λS(n)
η
on the space XS
0+
by
λS(n),f
η
=
π∈Πcus(n)∗
L(1/2,π) L(1/2,π η)
N(n) L(1,π;Ad)
f(νπ,S), f Cc
0
(XS
0+
),
where Πcus(n) is the set of all π Πcus(n) with conductor n, L(s, π) is the standard
L-function of π ([27]) and L(s, π; Ad) the adjoint square L-function of π ([11]).
We remark that, by the Rankin-Selberg method, the special value L(1,π;Ad) is
identified with the Petersson norm squared of the newform ϕπ new of π up to an
elementary positive constant. Thus, by non-negativity of the central L-values ([17],
[56]), the measure λS(n)
η
is non-negative. On the other hand, the space XS
0+
carries
a measure dλS
η
=
4DF/2 3
L(1,η)
v∈S
dλvv
η
, where DF is the absolute discriminant
of F , L(1,η) is the completed Hecke L-series of η and dλvv η is a measure on Xv0+
supported on the purely imaginary points, on which it is given by
dλvv
η
(iy) =
L(1/2,Iv(||v
iy/2
)) L(1/2,Iv(||v
iy/2
) ηv)
L(1,Iv(||v
iy/2
), Ad)
ζF,v(2)
L(1,ηv)
dPv(y) (1.2)
with
dPv(y) =



⎪(1





+ qv
−1)
log qv

1 qv
−iy
1 qv
−(1+iy)
2
dy, v Sfin,
1

Γ((1 + iy)/2)
Γ(iy/2)
2
dy, v Σ∞
(1.3)
the Kv-spherical Plancherel measure of PGL(2,Fv). Here, L(s, πv) denotes the
local L-factor of an irreducible smooth representation πv of GL(2,Fv) and L(1,ηv)
denotes Tates’ local L-factor of ηv.
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