120 J .W. COGDELL, £-FUNCTIONS FOR GLn
This allows us to obtain the analytic properties of the Eisenstein series from the
Poisson summation formula for 8q,, namely
8q,(a, g) = L <P(a~g) = L <Pa, 9 (0
= [a[-n [ det(g)[-^1 8\l>(a-^1 /g-^1 )
where the Fourier transform <I? on S(An) is defined by
<i?(x) = r <J?(y)'lfJ(ytx) dy.
}AxThis allows us to write the Eisenstein series as
E(g, <I?' s, 77) = I det(g) [^8 r e~ (a, g) [a [n^8 77( a) dx a
Jlal"?.l
+I det(g)[s-^1 f e~(a,t 9 -^1 )ta1n<^1 -sl 77 -^1 (a) dxa + J(s)
Jlal"?.lwhere
<5(s) = {~c<J?(O) I det(g)ls + c<f?(O) I det(g)j~-^1
s + tcr s-l+tcrif 77 is ramified
if 77(a) = [a[incr with CJ E IRwith c a non-zero constant. l,From this we derive easily the basic properties of our
Eisenstein series [45, Section 4].Proposition 2.1. The Eisenstein series E(g, <I?; s, 77) has a meromorphic continua-
tion to all of <C with at most simple poles at s = -iCJ, 1 - iCJ when 77 is unramified
of the form 77( a) = [a[incr. As a function of g it is smooth of moderate growth and
as a function of s it is bounded in vertical strips (away from t he possible poles),
uniformly for g in compact sets. Moreover, we have the functional equation
E(g, <I?; s, 77) = E(g\ <f?; 1 - s , 77-^1 )
where g' = tg-^1.Note that under t he center t he Eisenstein series transforms by the central char-
acter 77-^1.2.3.2. The global integrals
Now let us return to our Eulerian integrals. Let 7r and n' be our irreducible cuspidal
representations. Let their central characters be w and w'. Set 77 = ww'. Then for
each pair of cusp forms cp E Vn and cp' E Vn' and each Schwartz-Bruhat function
<I? E S( An) set
I(s; cp, cp', <J?) = { cp(g)cp'(g)E(g, <I?; s, 77) dg.
lzn(A) GLn(k)\ GLn (A)
Since the two cusp forms are rapidly decreasing on Zn (A) GLn(k)\ GLn(A) and the
Eisenstein is only of moderate growth, we see that the integral converges absolutely
for all s away from the poles of the Eisenstein series and is hence meromorphic. It