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

4π

1 − qv

−iy

1 − qv

−(1+iy)

2

dy, v ∈ Sfin,

1

4π

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.