1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. EULERIAN INTEGRALS FOR GLn 117

From this point, the fact that the integrals are Eulerian is a consequence of

the uniqueness of the Whittaker model for GLn. Take <p a smooth cusp form in

a cuspidal representation 7r of GLn(A). Assume in addition that <pis factorizable,

i.e., in the decomposition 7r = &/7rv of 7r into a restricted tensor product of local

representations, <p = ®<pv is a pure tensor. Then as we have seen there is a choice

oflocal Whittaker models so that Wcp(g) =IT WcpJ9v)· Similarly for decomposable

cp' we have the factorization w~, ( h) = IT w~~ ( hv).

If we substitute these factorizations into our integral expression, then since the

domain of integration factors Nm(A)\ GLm(A) =IT Nm(kv)\ GLm(kv) we see that

our integral factors into a product of local integrals

w(s;Wcp, w~,)

=II r w'Pv (~v In~m) w~~ (hv)I det(hv)l~-(n-m)/^2 dhv.

v }Nm(kv)\ GLm(kv)

If we denote the local integrals by

Wv(s;W'Pv' W~J

= r w'Pv (~v

}Nm(kv)\ GLm(kv)

InO -m ) W' c.p~ (h v )I det(h v v )ls-(n-m)/^2 dh v'

which converges for Re(s) >> 0 by the gauge estimate of [40, Prop. 2.3.6], we see
that we now have a family of Eulerian integrals.
Now let us return to the question of a functional equation. As in the case of
GL 2 , the functional equation is essentially a consequence of the existence of the
outer automorphism g r---t i(g) = g• = tg-^1 of GLn· If we define the action of this
automorphism on automorphic forms by setting 0(9) = cp(g•) = cp( Wng•) and let

iP~ = i o IP'~ o i then our integrals naturally satisfy the functional equation

I( s; <p, cp') = l(l - s; 0, 0')

where

f(s; cp, cp') = { iP~cp (h

jGLm(k)\ GLm(A)^1

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

We have established the following result.

Theorem 2.1. Let <p E Vn be a cusp form on GLn(A) and cp' E Vn' a cusp form on
GLm(A) with m < n. Then the family of integrals I(s; <p, cp') define entire functions
of s, bounded in vertical strips, and satisfy the functional equation

I(s; <p, cp') = f(1 - s; 0, 0').

Moreover the integrals are Eulerian and if <p and <p^1 are factorizable, we have

I(s; <p, cp') =II Wv(s; w'Pv' W~J

v

with convergence absolute and uniform for Re(s) >> 0.

The integrals occurring in the right hand side of our functional equation are
again Eulerian. One can unfold the definitions to find first that

l(l-s;0,0') = ~(1-s;p(wn,m)Wcp, W~,)