18 2. PRELIMINARIES

Since G(1) = DF

−1/2

, this completes the proof of the second statement.

From Lemma 2.2 and (2.17), Q(πv) ∈ (−1, 1). Combining this with the non-

negativity of L(1/2,π) L(1/2,π ⊗ η) proved in [17], we have the second statement.

Corollary 2.7. Let π be as in Lemma 2.6. Let η be a unitary idele-class

character of F × satisfying the following condition together with (2.18), (2.19) and

(2.20).

ηv(

v

) = −1 for any v ∈ S(n). (2.22)

Then,

Pη(π;

K0(n)) is zero unless fπ = n, in which case it equals

DF

−1/2

G(η)

L(1/2,π) L(1/2,π ⊗ η)

ϕπ new 2

.

2.7. Eisenstein series and their L-functions

For any unitary idele-class character χ and for any ν ∈ C, let I(χ||A

ν/2

) be the

space of all the smooth complex valued functions f on GA such that

f

(

t1 x

0 t2

g

)

= χ(t1/t2)

|t1/t2|Aν+1)/2 (

f(g) for all

t1 x

0 t2

∈ BA.

The group GA acts on I(χ||A

ν/2

) by the right translation. Then, the hermitian

pairing

f|f =

K

f(k)

¯

f (k) dk, f ∈ I(χ||A

ν/2

), f ∈ I(χ||A

−¯ ν/ 2

) (2.23)

is GA-invariant. For each place v, we have defined a local counterpart Iv(χv||v

ν/2

)

in 2.5.1. In the context of Eisenstein series, we always consider the Gv-hermitian

paring between Iv(χv||v

ν/2

) and

Iv(χv||−¯ ν/ 2

v

) similarly defined as (2.23) by integra-

tion on Kv, with respect to which Iv(χv||v

ν/2

) is unitary if ν ∈ iR. The global space

I(χ||A

ν/2

) is identified with the restricted tensor product of Iv(χv||v

ν/2

) with respect

to the family of Kv-invariant unit vectors f0,vv

(χ | |v

ν/2)

abbreviated to f0,χv

(ν)

for any

v ∈ Σ∞ ∪ S(fχ). We use similar abbreviation f1,χv

(ν)

for

f1,vv| (χ |v

ν/2)

and set

˜(ν)

f

1,χv

= χv(

v

)qv

−ν/2(1

+ qv

−1)

L(ν + 1,χv)

2

f1,χv

(ν)

. (2.24)

Lemma 2.8. Let v ∈ Σ∞ ∪ S(fχ). If ν ∈ iR, the functions f0,χv

(ν)

and

˜(ν)

f

1,χv

comprise an orthonormal basis of the 2 dimensional space Iv(χv||v

ν/2

)K0(pv).

The

local intertwining operator Mv(ν) : Iv(χv||v

ν/2

) −→ Iv(χv −1||v

−ν/2

),

[Mv(ν)fvν)](gv) (

=

Fv

fvν) (

(w0 [

1 x

0 1

] gv) dx

with suﬃciently large Re(ν) is computed on Iv(χv||v

ν/2

)K0(pv) by the basis {f0,χv

(ν)

,

˜(ν)

f

1,χv

}

as

Mv(ν)f0,χv

(ν)

= qv

−dv/2

Lv(ν, χv)

2

L(ν + 1,χv) 2

f

(−ν)

0,χv

−1

, (2.25)

Mv(ν)

˜(ν)

f

1,χv

= χv(

v

)2qv −ν−dv/2

Lv(ν, χv)

2

L(1 − ν, χv

−2)

˜(−ν)

f

1,χv

−1

. (2.26)