LECTURE 4. GLOBAL £-FUNCTIONS 139However, by the local functional equations, for each v ES we have_ (l _. ( )W W') _ ~(1 -s; p(wn,m)Wv, W~)
ev s, p Wn,m v, v - L(l -S,7fX7f -,)= w' (-l)n-1€(s 7f X 7f'. 1. ) \Il(s; Wv, W~)
v , v v, 'Pv L( S,7f X 7f ')
= w~(-l)n-le(s, 1fv X 7f~, 'l/Jv)ev(s, Wv, W~)
Combining these, we haveL(s, 7f x 7r^1 ) = (IT w~(-1)n-^1 e(s, 1fv x 7f~, 'l/Jv)) L(l - s, 'if x 'if').
vES
Now, for v tj. S we know that 7f~ is unramified, so w~ ( -1) = 1, and also that
e(s, 1fv x 7f~, 'l/Jv) = l. ThereforeIT w~(-l)n-le(s, 1fv x 7f~, 'l/Jv) =IT w~(-1r-^1 e(s, 1fv x 7f~, 'l/Jv)
vES v
= w^1 (-1r-^1 e(s, 7f x 7f^1 )
= e(S,7f X 7r^1 )
and we indeed have
L(s, 7f x 7r^1 ) = e(s, 7f x 7r^1 )L(l - s, 'if x 'if').
Note that this implies that e(s, 7f x 7r^1 ) is independent of 'ljJ as well.
Let us now turn to the boundedness in vertical strips. For the global integrals
I(s; r.p, r.p') or I(s; r.p, r.p, ) this simply follows from the absolute convergence. For
the £-function itself, the paradigm is the following. For every finite place v E S we
·know that there is a choice of Wv,i, W~,i' and v,i if necessary such that
L(s, 1fv x 7f~) = L \Il(s; Wv,i, w~'i) orL(s, 1fv x 7f~) = L \Il(s; Wv,i, w~'i> <I>v,i)·
If m = n-1 or m = n then by Theorem 3.8 we know that we have similar statements
for v E S=. Hence if m = n - 1 or m = n there are global choices 'Pi, r.p~, and if
necessary <I>i such thatL(s,1f x 7r^1 ) = LI(s;r.pi,'PD or L(s,1f x 7r^1 ) = Ll(s;r.pi,r.p~,<I>i)·
Then the boundedness in vertical strips for the £-functions follows from that of the
global integrals.
However, if m < n - 1 then all we know at those v E S= is that there is a
function Wv E W(7rv 0 7f~, 'l/Jv) = W(7rv, 'l/Jv) 0 W(7r~, 'l/J;^1 ) or a finite collection of
such functions Wv,i and of <I>v,i such thatL(s, 1fv X 7r~) = I(s; Wv) or L(s, 1fv X 7r~) = L I(s; Wv,i, <I>v,i)·
To make the above paradigm work form < n-1 we would need to rework the theory
of global Eulerian integrals for cusp forms in V.,,. 0 v.,,.,. This is naturally the space
of smooth vectors in an irreducible unitary cuspidal representation of GLn(A) x
GLm(A). So we would need extend the global theory of integrals parallel to Jacquet
and Shalika's extension of the local integrals in the archimedean theory. There