16 2. PRELIMINARIES

Proof. From assumptions, we have

(1/2,πv,ψF,v) v(1/2,πv ⊗ ηv,ψF,v) =

⎧

⎪ηv(−1),

⎨

⎪

⎩

(v ∈ Σ∞ ∪ S(fη)),

ηv(

v

), (v ∈ S(fπ)),

1, (v ∈ Σfin − S(fπ fη).

Thus, (1/2,π) (1/2,π ⊗ η) = ˜(fπ) η = ±1. Since η2 = 1, the central character of

π ⊗ η is also trivial. Hence the claim is a consequence of the functional equation

relating the values at s and 1 − s of L(s, π) L(s, π ⊗ η).

2.6.2. Fix an oF -ideal n such that n is square free. Then, the product K0(n) =

v∈Σfin

K0(nov), is an open compact subgroup of Kfin. Suppose Vπ

K0(n)K∞

= {0}

from now on. Then, Vπv

K0(nov)

= {0} for all v ∈ Σfin. This implies that fπ divides n.

For any ideal c dividing nfπ −1, let ϕπ,c be the function in Vπ corresponding to the

decomposable tensor

{

v∈S(c)

φ1,v} ⊗ {

v∈S(c)

φ0,v}

under the identification Vπ

∼

=

v

Vπv . The function ϕπ,oF is called the newform of

π and is denoted by ϕπ new.

Lemma 2.4. The functions ϕπ,c (nfπ

−1

⊂ c) form an orthogonal basis of the

invariant part Vπ

K0(n)K∞

, and

ϕπ,c 2

ϕπ new 2

=

v∈S(c)

{1 −

Q(πv)2}.

(2.17)

Proof. Since π is irreducible, there exists a positive constant C such that the

L2-inner

product on Vπ coincides with C · {

v

( | )v} by the isomorphism Vπ

∼

=

v

Vπv . Since c is relatively prime to fπ,

ϕπ,c

2

= C

v∈S(c)

{1 −

Q(πv)2}

by Lemma 2.2. Taking c = oF , we obtain C = ϕπ,oF

2.

This shows (2.17).

Let η be a unitary idele-class character of F

×

such that

η2

= 1, (2.18)

ηv(−1) = 1 for any v ∈ Σ∞, (2.19)

fη is relatively prime to n and ˜(n) η = 1. (2.20)

Instead of the zeta integral (2.16), which vanishes identically for ϕ ∈ Vπ

K0(n)K∞

unless η = 1, we consider

Z∗(s,

η; ϕ) = η(xη)

∗

Z

(

s, η; π

(

1 xη

0 1

)

ϕ

)

, ϕ ∈ Vπ

K0(n)K∞

,

where xη

(resp. xη) ∗ is the adele (resp. idele) whose v-component is

−f(ηv)

v

or

0 (resp. 1) according to v ∈ Σfin or v ∈ Σ∞, respectively. The factor η(xη) is

included to make the expression independent of the choices of uniformizers v.

Lemma 2.5. Let η be as above. Then, for any c dividing nfπ −1, we have

Z∗(s,

η; ϕπ,c) =

DF−1/2 s

G(η) {

v∈S(c)

ηv(

v

)qv/2−s 1

− Q(πv) } L(s, π ⊗ η).