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 = 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
| |v
ν/2)
abbreviated to f0,χv
(ν)
for any
v Σ∞ S(fχ). We use similar abbreviation f1,χv
(ν)
for
f1,vv| |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)
Previous Page Next Page