2.7. EISENSTEIN SERIES AND THEIR L-FUNCTIONS 19
Proof. If ν iR, the hermitian inner product on Iv(χv||v
ν/2
) is Gv-invariant.
Using the easily confirmed relation
1 Q(Iv(χv||v
ν/2))2
= χv(
v
)qv
−ν/2(1
+ qv
−1)
L(ν + 1,χv)
2
−2
, ν iR,
we infer the first statement from Lemma 2.2. The formula (2.25) is proved in
[5, Proposition 4.6.7] when dv = 0; the general case is similar. By (2.9), we see
that Mv(ν) sends the vector f1,χv
(ν)
to the vector f
(−ν)
1,χv
−1
multiplied by the same
constant occurring in (2.25). Then, taking the modification factor in (2.24) into
account, we obtain (2.26).
2.7.1. Fix χ. A family of smooth functions f
(ν)
I(χ||A
ν/2
) with varying
ν C is said to be a flat section if f
(ν)|K
is independent of ν. A flat section
f
(ν)
I(χ||A
ν/2
) is K-finite if R(K)f
(0)
spans a finite dimensional space in I(χ).
For such f
(ν),
the Eisenstein series is initially defined by the absolutely convergent
series
E(f
(ν);
g) =
γ∈BF \GF
f
(ν)(γg),
g GA, Re(ν) 1.
For each g GA, the holomorphic function ν E(f
(ν);
g) on Re(ν) 1 is contin-
ued meromorphically to the whole complex plane, holomorphic on the imaginary
axis. The unique singularity of E(f
(ν);
g) on the half-plane Re(ν) 0 is a possible
pole at ν = 1, which occurs only when
χ2
= 1.
2.7.2. Let n be a square free oF -ideal. Then, the invariant part I(χ||A
ν/2
)K0(n)K∞
is non zero only when = oF ; we assume this from now on. For each oF -ideal c di-
viding n, let
fχ,c)
I(χ||A
ν/2
) be the function coming from the decomposable tensor
{
v∈S(c)
˜(ν)
f
1,χv
}⊗{
v∈S(c)
f0,χv
(ν)
} by the isomorphism I(χ||A
ν/2
)

=
v
Iv(χv||v
ν/2
).
Lemma 2.9. The family of functions
fχ,c)
I(χ||A
ν/2
) C) is flat. When
ν iR, the set of functions
{fχ,c)|
n c } is an orthonormal basis of the invariant
part I(χ||A
ν/2
)K0(n)K∞
.
Proof. For any place v, the restriction f0,χv
(ν)
|Kv is identically 1. If v Σfin,
it turns out that
˜(ν)
f
1,χv
(k) equals
qv/2 1
or −qv
−1/2
according to k K0(pv) or k
Kv K0(pv), respectively. From these observations, the first assertion is evident.
Since the inner product (2.23) is a tensor product of inner products on local
spaces Iv(χv||v
ν/2
) which contains a unit vector
f0,v)
at all places v, the second
assertion follows from the first statement of Lemma 2.8.
From now on, we abbreviate
E(fχ,c);
g) to Eχ,c(ν; g). Let
Eχ,c(ν;

g) =
F \A
Eχ,c(ν;[ 1 x
0 1
] g) dx
be the constant term of Eχ,c(ν; g). Then,
Lemma 2.10.
Eχ,c(ν;

g) =
fχ,c)(g)
+ DF
−1/2
Aχ,c(ν)
L(ν,
χ2)
L(ν + 1,χ2)
f
(−ν)
χ−1,c
(g), g GA,
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