2.6. CUSPIDAL AUTOMORPHIC REPRESENTATIONS AND THEIR L-FUNCTIONS 15

corresponds to the vector (1 − qv

−2)−1{f0,vv) (χ

− (−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) (ν

such that

f0,v)(e) (ν

= 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). (ν

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 R× 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 Vπ 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 fπ 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 fπ 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 Vπ

∼

=

v

Vπv is

made so that if ϕ ∈ Vπ 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 fπ is square free and π is K∞-spherical. Let η be

an idele-class character of F × such that η2 = 1 and such that fη is relatively prime

to fπ. Then,

L(1/2,π) L(1/2,π ⊗ η) = 0 unless ˜(fπ) η = 1.