138 J.W. COGDELL, £-FUNCTIONS FOR GLn
Theorem 4.1. The global £ - functions L(s, -rr x -rr') are nice in the sense that
(1) L(s, -rr x -rr') has a meromorphic continuation to all of C,
(2) the extended function is bounded in vertical strips (away from its poles),
(3) they satisfy the functional equationL(s, -rr x -rr') = c:(s, 7r x 7r^1 )L(l - s, if x if').
To do so, we relate the £-functions to the global integrals.
Let us begin with continuation. In the case m < n for every cp E V71" and
cp' E V71"' we know the integral I(s; cp, cp') converges absolutely for all s. From the
unfolding in Lecture 2 and the lo cal calculation of Lecture 3 we know that for
Re( s) > > 0 and for appropriate choices of cp and cp' we have
vWe know that each ev(s; Wv, W~) is entire. Hence L(s, -rr x -rr') has a meromorphic
continuation. If m = n then for appropriate cp E V71", cp' E V71"'' and E S(An) we
again have
I(s; cp, cp', <I>)= (rr ev(s; W"'v' W~~' <I>v)) L(s, 7r x -rr').
vESOnce again, since each ev(s; Wv, W~, v) is entire, L(s, 7r x 7r^1 ) has a meromorphic
continuation.
Let us next turn to the functional equation. This will follow from the functional
equation for the global integrals and the local functional equations. We will consider
only the case where m < n since the other case is entirely analogous. The functional
equation for the global integrals is simply
I(s; cp, cp') = 1(1 - s; ij5, ij5').
Once again we have for appropriate cp and cp'I(s; cp, cp') = (rr ev(s; W"'v' W~J) L(s, 7r x 7r
1
)
vESwhile on the other side
1(1-s;ij5,ij5') =(IT ev(l-s;p(wn,m)W"'v' W~J) L(l - s,if x if').
vES