LECTURE 3. FUNCTIONAL EQUATIONS AND MULTIPLICATIVITY 319£-function and root number
£ - functions are now defined using 1 - functions. When O" is tempered, we define
L(s, O", ri) as the inverse of the normalized polynomial P(q-s) in q-s satisfying
P(O) = 1 and
(3.16) 1(s, O", ri, 'l/JF) =c(s, O", ri, 'lj;p )L(l - s, O", f)/ L(s, O", ri)·
The £ - function L(s, O", ri) and the root number c(s, O", ri , 'lj;p) are also uniquely
defined by (3.16). To proceed we need the following conjecture whose validity is
now proved in almost all the cases [Shl,As,CSh,Kl].
Conjecture 3.10 [Shl]. Suppose O" is tempered. Then L(s, O", ri) are all holomor-
phic for Re(s) > 0.
Under this conjecture, L(s, O", ri) are now also multiplicative if O" is tempered.
(See below.)
To define L(s, O", ri) for any irreducible x - generic representation, we appeal to
Langlands classification [La3,Si]. We embed O" c Ind O"~ 0 1, where O"~
M'(N'nM)iM
is quasi-tempered with a negative Langlands parameter LI. Then O"b is tempered.
By multiplicativity, we then write 1(s, O", ri, 'l/JF) as a product of appropriate 1-
functions 1( s, O"~,j, rij, 'I/; F), where j runs over a finite index set determined by M'
and M, i.e.,
(3.17) 1(s,O",ri,'l/JF) =II 1(s,O"~,j,rij,'l/;F)·
j
More precisely, for each j, there exist Levi subgroups (not necessarily maximal)
Mj and M j of G with T C Mj C Mi as a maximal Levi subgroup. The repre-
sentation O"~,i is a quasi-tempered representation of Mj for which O"b,j is tempered.
The representation rij of L Mj is an irreducible constitutent of the action of L Mj on
the Lie algebra of the £ - group of Mi n Nj, where Nj c U is the uni potent radical
of Pj = MjNj. Thus the 1 - function 1(s, O"~,i' rij, 'l/JF) is a 1 - function attached to
the pair (Mj, Mj).
When LI = 0, by Conjecture 3.10, the product of the numerators of
1(s, O"b,j, rij, 'l/JF) equ als to the numerator of the product and if L(s, O"b,j, r ij)-^1
denotes the normalized numerator of 1(s, O"b,j, rii> 'l/JF ), i.e., the reciprocal of the
tempered £-function attached to O"b,j and r ij by means of the pair (Mj, Mj), we
then use L(s,O"~,j,rij) to denote its analytic continuation to LI. We now set(3.18) L(s,O",ri ) =II L(s,O"~,j,rij)·
jThis agrees with the way Artin £-functions are defined [La3,KSh,Shl,T]. Details
are given in [Sh3]; also see the discussion in pages 862 and 863 of [KS2]. The root
number is then defined uniquely to satisfy (3.16). We should point out that in
Definition (3.18) we do not need to assume the validity of Conjecture 3.10. But if
valid, it implies that if the representations are given by their Langlands parameters,
then the £ - functions are multiplicative on their inducing quasi-tempered data.
This generalizes Proposition 9.4 of [JPSSl] to our general setting.
Having defined our £ - functions and root numbers everywhere, we set
(3.19) L(s,Jr,ri) =II L(s,7rv,ri)
v