10 2. PRELIMINARIES

Haar measure

d×xv

on Fv

×

by

d×xv

= cv dxv/|xv|v, where cv = 1 if v ∈ Σ∞ and

cv = (1 − qv

−1)−1

if v ∈ Σfin. We fix a Haar measure of the idele group

A×

by

d×x

=

v

d×xv.

For y 0, let y ∈

A×

be the idele such that y

ι

=

y1/dF

for

all ι ∈ Σ∞ and y

v

= 1 for all v ∈ Σfin. Then, y → y is a section of the idele

norm | |A :

A×

→ R+,

∗

which allows us to identify

A×

with the direct product of

R+

×

= {y| y 0 } and the norm one subgroup

A1

= {x ∈

A×||x|A

= 1 }. We fix a

Haar measure

d1u

on

A1

by requiring

A×

f(x)

d×x

=

+∞

0 A1

f(y u)

d×y d1u,

f ∈

L1(A×).

We fix Haar measures dhv, dnv, dkv and dgv on groups Hv, Nv, Kv and Gv

respectively by requiring vol(Kv;dkv) = 1 and

dhv =

d×t1,v d×t2,v

if hv =

t1,v 0

0 t2,v

, t1,v, t2,v ∈ Fv

×

,

dnv = dxv, if nv = 1 xv

0 1

, xv ∈ Fv,

dgv = dhv dnv dkv if gv = hv nv kv (hv ∈ Hv, nv ∈ Nv, kv ∈ Kv).

The adele group GA is decomposed to a direct product of GA

1

= {g ∈ GA|| det g|A

= 1 } and A = {y 12| y 0 }. By taking the tensor product of measures dgv on

Gv, we fix a Haar measure dg on GA; then the decomposition GA = GA 1 A yields

the Haar measure d1g on GA 1 such that dg = d1g d×y. By virtue of the Iwasawa

decomposition,

G1

A

ϕ(g)

d1g

=

+∞

0 x∈A

(u1,u2)∈(A1)2

k∈K

ϕ

1 x

0 1

tu1 0

0

t−1u2

k

t−2 d×t d1u1d1u2

dx dk,

(2.1)

where dk is the Haar measure on K with total mass 1.

2.3.1. Let g∞ be the complexification of the Lie algebra of G∞. The universal

enveloping algebra of the complexification of a real Lie algebra l is denoted by U(l)

and the center of U(l) is denoted by Z(l). Let g∞ and gι denote the Lie algebras of

G∞ and Gι, respectively. Then, U(g∞)

∼

=

ι∈Σ∞

U(gι) and Z(g∞)

∼

=

ι∈Σ∞

Z(gι).

For any function ϕ on GA, the right translate of ϕ by an element g ∈ GA is denoted

by R(g)ϕ, i.e., [R(g)ϕ](x) = ϕ(xg). For a smooth function ϕ on GA, its right-

derivation by a differential operator D ∈ U(g∞) is also denoted by R(D)ϕ.

2.3.2. Let v ∈ Σfin. The Hecke algebra for the pair (Gv, Kv) is denoted by

H(Gv; Kv). Recall that it consists of all the C-valued compactly supported func-

tions on Gv constant on each Kv-double coset; it has the unit given by 1Kv =

vol(Kv;dgv)−1

χKv . We define an element Tv ∈ H(Gv, Kv) by

Tv =

1

vol(Kv;dgv)

χKv[

v

1

]Kv

.

2.4. L-functions for idele-class characters

We refer to [49], [42].