1549380232-Automorphic_Forms_and_Applications__Sarnak_

150 J.W. COGDELL, £-FUNCTIONS FOR GLn

5.2. Inverting the integral representation

Let IT b e as above and let ~ E Vn be a decomposable vector in the space Vn of

IT. Since IT is generic, then fixing local \iVhittaker models W(ITv, 'l/Jv) at all places,

compatibly normalized at the unramified places, we can associate to ~ a non-zero

function Wi;(g) =IT Wi;v (gv) on GLn(A) which transforms by the global character

'l/J under left translation by Nn(A), i.e., Wi;(ng) = 'lfJ(n)Wi;(g). Since 'l/J is trivial on

rational points, we see that Wi;(g) is left invariant under Nn(k). We would like to

use WE to construct an embedding of Vn into the space of (smooth) automorphic

forms on GLn(A). The simplest idea is to average Wi; over Nn(k)\ GLn(k), but

this will not b e convergent. However, if we average over the rational points of t he

mirabolic P = P n then the sum

Ui;(g) = WE(pg)

N n(k)\ P(k)

is absolutely convergent. For t h e relevant growth properties of Ui; see [9]. Since IT

is assumed to have automorphic central character, we see that Ui; (g) is left invariant

under both P ( k) and the center Zn ( k).

Suppose now that we know that L(s, IT x 7r^1 ) is nice for all 7r^1 E T(m). Then we

will hope to obtain the remaining invariance of Ui; from the GLn x GLm functional

equation by inverting the integral representation for L(s, IT x 7r^1 ). With this in

mind, let Q = Qm be the miraboli c subgroup of GLn which stabilizes the standard

unit vector tem+l, t hat is the column vector all of whose entries a re 0 except the

(m + l)th, which is 1. Note t ha t if m = n - 1 then Q is nothing more t h a n t he

opposite mirabolic P = t p-l to P. If we let <Xm be the permutation matrix in

GLn(k) given by

then Qm = a;;-,^1 cxn-1Pa;;:~ 1 cxm is a conjugate of P and for a ny m we have that P(k)

and Q(k) generate all of GLn(k). So now set

Vi;(g) =

N'(k)\ Q(k)

where N' = a;;-,^1 Nn <Xm C Q. This sum is again absolutely conver gent and is invari-

ant on the left by Q(k) and Z(k). Thus, to emb ed IT into the space of automorphic

forms it suffices to show Ui; = Vi;, for then we get invarian ce of Ui; under all of

GLn(k). It is this t hat we will attempt to do usi ng the integral representations.

Now let ( 7r^1 , V.,,.,) be an irreducible subrepresentation of the space of automor-

phic forms on GLm(A) and assume cp' E V,,., is also factorizable. Let

I(s; Ui;, cp') = ( lP~Ui; (h

1

) cp'(h)I det(h)ls-^1 /^2 dh.

jGLm(k)\ GLm (A)

This integral is always absolutely convergent for R e(s) >> 0, and for alls if ?r^1 is
cuspidal. As with the usual integral representation we h ave that this unfolds into
the Euler product