2.5. REPRESENTATIONS OF LOCAL GROUPS AND THEIR L-FACTORS 13

2.5. Representations of local groups and their L-factors

Let v be a place of F and πv an irreducible unitarizable smooth representation

of Gv. We always assume that πv is infinite dimensional, or equivalently generic;

let Vπv = W(πv,ψF,v) be the ψF,v-Whittaker realization of πv.

2.5.1. Let v ∈ Σfin. Then, the maximal compact subgroup Kv of Gv admits a

decreasing filtration by open compact subgroups

K0(pv

n)

= { a b

c d

∈ Kv| c ≡ 0 (mod pv

n)

}, n ∈ N.

From [6], there exists a unique integer c(πv) ∈ N such that dimC Vπv

K0(pv

n)

= sup(n−

c(πv) + 1, 0) for any n ∈ N. For any quasi-character ηv of Fv

×,

the integral

Zv(s, ηv; φv) =

Fv×

φv,0 ([ t 0

0 1

]) ηv(t)

|t|v−1/2 s d×t,

φv ∈ Vπv (2.7)

is called the local zeta integral. By the theory of local new forms (cf. [44, §1]),

there exists a unique element φ0,v ∈ Vπv

K0(pv(πv)) c

such that

Z(s, ηv; φ0,v) = vol(ov

×;d×t)

ηv(

v)−dv qvv(s−1/2) d

L(s, πv ⊗ ηv), Re(s) 0

(2.8)

for any unitary unramified character ηv of Fv ×.

Let us recall the construction of unramified principal series. For any unramified

quasi-character χv of Fv

×,

let Iv) (χv) be the space of all the smooth functions fv :

Gv → C satisfying fv

(

t1 x

0 t2

g = χv(t1/t2)

|t1/t2|v/2 1

fv(g) for any

t1 x

0 t2

∈ Bv.

We let the group Gv act on Iv(χv) by right translation. The space Iv(χv) contains

a unique Kv-invariant function

f0,vv) (χ

such that

f0,vv)(e) (χ

= 1. For our purpose, we

need information on the structure of the K0(pv)-fixed part of Iv(χv). To describe

it, set

f1,vv) (χ

= πv

−1

v

0

0 1

f0,vv) (χ

− Q(πv)

f0,vv) (χ

with Q(πv) =

χv(

v

) + χv(

v

)−1

qv/2 1

+ qv

−1/2

.

(2.9)

We define a Gv-intertwining operator Iv(χv) → W(Iv(χv),ψF,v) by mapping f ∈

Iv(χv) to the function φ ∈ W(Iv(χv),ψF,v) such that

φ(gv) = χv(

−dv

v

)

Fv

f

(

0 −1

1 0

[ 1 x

0 1

] gv

)

ψF,v(−xv) dxv, gv ∈ Gv, (2.10)

where the integral is interpreted as in [5, p.498, (6.9)]. Let φi,v (i = 0, 1) be

the image of (1 − χv(

v

)2

qv

−1)−1

fi,vv) (χ

in W(Iv(χv),ψF,v) under the intertwining

operator. Then, from [5, Proposition 4.6.8], we have

φ0,v

(

m

v

0

0 1

)

= δ(m + dv 0) qv

−(m+dv)/2

qvv(m+dv+1) ν

− qv

−νv(m+dv+1)

qvv

ν

− qv

−νv

, m ∈ Z

(2.11)

with qvv

ν

= χv( v). We remark that a small modification is necessary because dv

is not necessarily 0 in our case.