1549380232-Automorphic_Forms_and_Applications__Sarnak_

116 J.W. COGDELL, £-FUNCTIONS FOR GLn

2.2.2. The global integrals

We now have the prerequisites for writing down a family of Eulerian integrals for

cusp forms <p on GLn twisted by automorphic forms on GLm form< n. Let <p E V1T

be a cusp form on GLn(A) and <p^1 E V1T' a cusp form on GLm(A). (Actually, we

could take <p^1 to be an arbitrary automorphic form on GLm(A).) Consider the

integrals

I(s; <p, <p^1 ) = 1 lP'~t.p (~ ~) t.p'(h)I det(h)ls-^1 /^2 dh.

GLm(k)\ GLm(A)

The integral I(s; <p, t.p') is absolutely convergent for all values of the complex para-

meter s, uniformly in compact subsets, since the cusp forms are rapidly decreasing.

Hence it is entire and bounded in any vertical strip as before.

Let us now investigate the Eulerian properties of these integrals. We first

replace lP'~<p by its Fourier expansion.

I(s;<p,<p^1 ) = 1 lP'~<p (~ IO ) t.p'(h)ldet(h)ls-^1 /^2 dh

GLm(k)\ GLm(A) n-m

= J L w~ ((6 1 ° ) (~ 1 ° ))

GLm(k)\ GLm(A) -yENm(k)\ GLm(k) n-m n-m

<p^1 (h)I det(h)ls-(n-m)/^2 dh.

Since <p'(h) is automorphic on GLm(A) and I det(1)I = 1 for ')' E GLm(k) we

may interchange the order of summation and integration for Re( s) > > 0 and then

recombine to obtain

I(s; <p, <p^1 ) = r w~ (h^0 ) <p^1 (h)I det(h)ls-(n-m)/^2 dh.

}Nm(k) \ GLm(A) Q In-m

This integral is absolutely convergent for Re(s) >> 0 by the gauge estimates of [40,

Section 13] and this justifies the interchange.

Let us now integrate first over Nm(k)\ Nm(A). Recall that for n E Nm(A) C

Nn(A) we have W~(ng) = 'l/i(n)W~(g). Hence we have

J(s;<p,t.p') = { { W~ ((n I 0 ) (h 0 ))

jNm(A)\ GLm(A) j Nm(k)\ Nm(A) Q n-m Q ln-m

<p^1 (nh) dn I det(h)ls-(n-m)/^2 dh

= l m(A)\ GLm(A) W~ (~ ln~m)

f 'l/i(n)t.p'(nh) dn I det(h)ls- cn-mi1^2 dh

}Nm(k)\ Nm(A)

= f w~ (h o ) w~,(h)I det(h)ls-cn-mJ/^2 dh

}Nm(A)\ GLm(A) Q ln-m

= w(s; w~, w~,)

where W~,(h) is the '¢'-^1 -Whittaker function on GLm(A) associated to <p^1 , i.e.,

w~,(h) = r t.p^1 (nh)'¢'(n) dn,

}Nm(k)\ N,,,(A)

and we retain absolute convergence for Re(s) >> 0.