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)
such that
f0,vv)(e)
= 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)
= πv
−1
v
0
0 1
f0,vv)
Q(πv)
f0,vv)
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)
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.
Previous Page Next Page