14 2. PRELIMINARIES

Lemma 2.2. Suppose Iv(χv) is irreducible and unitarizable. Then, the number

Q(πv) is real. The vectors

f0,vv) (χ

and

f1,vv) (χ

yield a basis of the two dimensional

space Vπv

K0(pv)

such that

(f1,vv)|f0,vv (χ (χ )

) = 0, f1,vv

(χ )

2

= {1 −

Q(πv)2}f0,vv

(χ )

2,

(2.12)

for any Gv-invariant hermitian inner product ( | ) on Iv(χv). Let ηv be a unitary

unramified character of Fv

×.

Then, for suﬃciently large Re(s), we have

Zv(s, η; φ0,v) = vol(ov

×;d×t)

ηv(

v)−dv

qvv

d (s−1/2)

L(s, Iv(χv)), (2.13)

Zv(s, ηv; φ1,v) = (ηv(

v

)qv/2−s 1

− Q(πv)) Zv(s, η; φ0,v). (2.14)

Proof. In this proof, we abbreviate fj,vv

(χ )

to fj,v. Let Tv be as in 2.3.2. From

[5, Proposition 4.6.6], the operator πv(Tv) acts on the space Vπv

Kv

by the scalar

qv/2(χv( 1

v

)+χv(

v

)−1). By the unitarity, we have the equation (πv(Tv)f0,v|f0,v) =

(f0,v|πv(Tv)f0,v), from which Q(πv) ∈ R is inferred. By the unitarity together with

the Kv-invariance of f0,v, we have

(2.15)

(f0,v|πv

−1

v

0

0 1

f0,v) = vol

(

Kv

v

0

0 1

Kv;dgv

)−1

(πv(Tv)f0,v|f0,v)

=

qv/2(χv( 1

v) + χv(

v)−1)

1 + qv

f0,v

2

.

Note that vol

(

Kv

v

0

0 1

Kv;dgv

)

= (1+qv) vol(Kv;dgv) from [5, p.494, Eq.(6.4)].

Now, by (2.15), the formula (2.12) is proved easily. In particular, f0,v and f1,v are

linearly independent. Using the relation πv

−1

v

0

0 1

f0,v ([ t 0

0 1

]) = f0,v tv

−1

0

0 1

,

the formula (2.14) is proved by a change of variable. When dv = 0, the formula

(2.13) is inferred from [5, Proposition 3.5.3] together with [5, p.358, Eq.(7.33)]; the

argument is easily extended to the case dv 0 with a minor modification.

For our purpose, we need only representations πv with c(πv) = 0 or 1:

• If c(πv) = 0, i.e., πv is Kv-spherical, then, πv

∼

=

Iv(||vv ν ) with

(qv

−νv

, qvv

ν

), (νv ∈ C/2πi(log

qv)−1Z),

the Satake parameter of πv. We have

L(s, πv ⊗ ηv) = (1 − ηv( v)qv

−(s+νv))−1(1

− ηv(

v)q−(s−νv))−1

for any unramified character ηv of Fv

×.

The local new form φ0,v corre-

sponds to the vector (1 − qv

−(1+2νv))−1 f0,vv) (χ

in Iv(χv) with χv = | |vv ν .

• If c(πv) = 1, then there exists

v

∈ {0, 1} such that πv is isomorphic

to the twist of the Steinberg representation Stv ⊗ sgnvv , or equivalently,

the unique proper subrepresentation of Iv(sgnvv |

|v/2), 1

where sgnv is the

unique unramified character of Fv

×

such that sgnv( v) = −1. We have

L(s, πv ⊗ ηv) = (1 − (−1)

v

ηv( v) qv

−(s+1/2))−1,

(1/2,πv ⊗ ηv,ψF,v) = −(−1)

v

ηv(

v

)

for any unramified character ηv of Fv

×.

By the intertwining operator

Iv(χv) → W(Iv(χv),ψF,v) defined by (2.10), the local new form φ0,v of πv