1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. EULERIAN INTEGRALS FOR GLn 113

In this way, to n and x we have associated a family of global Eulerian integrals

with nice analytic properties as well as for each place v a family of local integrals

convergent for Re(s) >> 0.

2.2. Eulerian integrals for GLn x GLm with m < n

Now let ( n, V11') be a cuspidal representation of G Ln (A) and ( n', V11'') a cuspidal

representation of GLm(A) with m < n. Take <p E V7l' and <p' E V7r'· At first blush, a

natural analogue of the integrals we considered for GL 2 with GL 1 twists would be

f <fl (h ) <p'(h)I det(h)ls-<n-ml/^2 dh.

}GL,,.(k)\ GLm(A) In-m

This family of integrals would have all t he nice analytic properties as before (entire

functions of finite order satisfying a functional equation), but they would not be

Eulerian except in the case m = n - 1 , which proceeds exactly as in the GL 2 case.

The problem is that the restriction of the form <p to GLm is too brutal to

allow a nice unfolding when the Fourier expansion of <p is inserted. Instead we will

introduce projection operators from cusp forms on GLn(A) to cuspidal functions

on Pm+i(A) which are given by part of the unipotent integration t hrough which

the Whittaker function is defined.

2.2.1. The projection operator

In GLn, let Y n,m be the unipotent radical of the standard parabolic subgroup

attached to the partition (m + 1, 1, ... , 1). If 'ljJ is our standard additive character

of k\A, then 'ljJ defines a character of Y n,m(A) trivial on Y n,m(k) since Y n,m C Nn.

The group Y n,m is normalized by GLm+l C GLn and t he mirabolic subgroup

P m+l C GLm+l is the stabilizer in GLm+l of the character 'lj;.

Definition. If <p(g) is a cusp form on GLn(A) define the projection operator IP'~

from cusp forms on GLn(A) to cuspidal functions on Pm+i(A) by

IP'~<p(p) =I det(p)J-(n-;i-1) r <p (y ( p In-m-1)) 'l/;-l(y) dy

}Yn,,,.(k)\ Yn,m(A)

As the integration is over a compact domain, the integral is absolutely conver-

gent. We first analyze the behavior on Pm+i(A).

Lemma 2.1. The function IP'~<p(p) is a cuspidal function on Pm+ 1 (A).

Proof: Let us let <p'(p) denote the non-normalized projection, i.e., for p E Pm+1(A)
set

<p^1 (p) =I det(p)J (n-;i-l )IP'~<p(p).

It suffices to show this fu nction is cuspidal. Since <p(g) was a smooth function on
GLn(A), <p'(p) will remain smooth on Pm+i(A). To see that <p'(p) is automorphic,
let 'YE Pm+I(k). Then

<p'('Yp) = j <p (y (Z ~) (~ ~)) 'l/;-l(y) dy.

Yn,m(k)\ Yn,m(A)