14 2. PRELIMINARIES
Lemma 2.2. Suppose Iv(χv) is irreducible and unitarizable. Then, the number
Q(πv) is real. The vectors
f0,vv)
and
f1,vv)
yield a basis of the two dimensional
space Vπv
K0(pv)
such that
(f1,vv)|f0,vv )
) = 0, f1,vv
)
2
= {1
Q(πv)2}f0,vv
)
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
)
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)
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
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