2.6. CUSPIDAL AUTOMORPHIC REPRESENTATIONS AND THEIR L-FUNCTIONS 15
corresponds to the vector (1 qv
−2)−1{f0,vv)
(−1)
v
πv
−1
v
0
0 1
f0,vv)}χ(
in Iv(χv) with χv = sgnvv |
|v/2.1
2.5.2. Let v Σ∞ and πv a Kv-spherical unitarizable (gv, Kv)-module. Then,
πv is isomorphic to the principal series Iv(||v) ν constructed in the same way as
the pv-adic case: it contains a unique Kv-fixed vector
f0,v)
such that
f0,v)(e)
= 1.
Then, the isomorphism πv

= Iv(||v)
ν
can be fixed in such a way that the local
new vector φ0,v is the element of the Whittaker model Vπv which corresponds to
ΓR(1+2ν)
f0,v).
We identify the Poincar´ e upper half plane H with the homogeneous
space SL2(R)/SO(2) by assigning the point (ai + b)/(ci + d) of H to a matrix
a b
c d
in SL2(R). Let Ωv be the Casimir element of Gv

=
GL(2, R) which corresponds to
(−2) times the hyperbolic laplacian −y2(∂2/∂x2 + ∂2/∂y2) on H. Then,
πv(Ωv)φ0,v =
(2ν)2 1
2
φ0,v,
and
L(s, πv ηv) = ΓR(s ν + a + ) ΓR(s + ν + a + )
for any character ηv(x) = |x|asgn(x) of with a iR, {0, 1}.
2.6. Cuspidal automorphic representations and their L-functions
Let (π, Vπ) be an irreducible cuspidal automorphic representation of GA with
trivial central character, where the representation space is contained in the sub-
space of K-finite functions in the
L2-space L2(GF
ZA\GA;dg). Let η be a unitary
idele-class character of F
×.
The global zeta integral of π η is defined by
Z(s, η; ϕ) =
F ×\A×
ϕ ([ t 0
0 1
]) η(t)
|t|s−1/2
A
d×t,
ϕ Vπ, (2.16)
which converges absolutely for any s C giving an entire function on C. The
conductor of π is, by definition, the oF -ideal such that fπov =
pv(πv) c
for all
v Σfin.
Fix a family {πv}v of irreducible smooth representations of Gv for all places v
such that π

=
v
πv.
2.6.1. Suppose is square free and π is K∞-spherical; thus πv is one of
representations recalled in 2.5.1 and 2.5.2. Let Vπv = W(πv,ψF,v) be the ψF,v-
Whittaker realization and φv,0 Vπv the local new vector. At almost all v, the
Whittaker function φ0,v is Kv-invariant and φv,0(e) = 1 ([5, Proposition 3.5.3]).
We fix a Gv-invariant hermitian inner product ( | )v on Vπv such that φ0,v 2 = 1,
once and for all. Furthermore, we assume that the identification

=
v
Vπv is
made so that if ϕ corresponds to the decomposable tensor
v
φv then the
global Whittaker function Wϕ(g) defined by the integral [5, Eq.(5.19)] coincides
with
v
φv(gv) for any g GA.
Lemma 2.3. Let π be an irreducible cuspidal automorphic representation with
trivial central character such that is square free and π is K∞-spherical. Let η be
an idele-class character of F × such that η2 = 1 and such that is relatively prime
to fπ. Then,
L(1/2,π) L(1/2,π η) = 0 unless ˜(fπ) η = 1.
Previous Page Next Page