LECTURE 3
Functional equations and rnultiplicativity
3.1. Local Coefficients, Non-constant Term and the Crude Func-
tional Equation
Changing the splitting in U , we may assume each Xv is defined by means of 7/Jv
through equation (1.5). At each v, let >-xJs, 1rv) be the Whittaker functional
defined by equation (1.11). Next, let A(s, 1rv, wo) be the local intertwining operator
defined by (2.10). Finally, let >-xJ-s, w 0 (7rv)) be the corresponding functional
defined for 1(-s, w 0 (7rv)) by means of (1.11). Using our assumption on Xv, Rodier's
theorem points to the existence of a complex function CxJs, 1rv) of s, depending
on 7r v, Xv and wo such that
(3.1)
This is what we call the "Local Coefficient" attached to s, 1rv, Xv and wo
(cf. [Sh2,Sh3]). The choice of w 0 is now specified by our fixed splitting as in [Sh4].
Now, let
(3.2) Ex(s,
lu(F)\U
the x-nonconstant term of E(s, Cf> 8 ,g, P) (cf. [Sh2,Sh3,Sh6]), where x = 0 vXv·
If we substitute for E(s, if> 8 , ug, P) its definition in (2.3) and do some telescop-
ing, we get, using orthogonality of x, that
(3.3)
v
where fv is the local component off defined by (2.6) in which <p = 0v<fJv is identified
with Wcp = 0 vWv, where Wcp is defined by (1.7). As explained in [SJ, Wv(ev) = 1
for almost all v. We now appeal to the following formula of Casselman-Shalika
[CS]:
Theorem 3.1 (Casselman-Shalika (CS]). Assume 1rv and 7/Jv are both unram-
ified and if fv(ev) defines a Whittaker function Wv in the Whittaker model of 1rv,
assume Wv ( ev) = 1. Observe that this is the case for almost all v. Then
rn
(3.4) Axv(S,1rv)(fv) =II L(l +is,7rv,'i)-^1.
i=l
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