1549380232-Automorphic_Forms_and_Applications__Sarnak_

114 J.W. COGDELL, £-FUNCTIONS FOR GLn

Since"( E Pm+ 1 (k) and Pm+l normalizes Yn,m and stabilizes 'ljJ we may make the

change of variable y f---+ ( 6 ~) y ( 6 ~)-l in this integral to obtain

cp'("fp) = J cp ((6 ~) y (~ ~)) 'lj;-l(y) dy.

Y n,m(k)\ Y n ,m (A)

Since cp(g) is automorphic on GLn(A) it is left invariant under GLn(k) and we find
that cp'("fp) = cp'(p) so that cp' is indeed automorphic on Pm+ 1 (A).
We next need to see that cp' is cuspidal on Pm+ 1 (A). To this end, let UC Pm+l
be the standard uni potent subgroup associated to the partition ( n 1 , ... , nr) of m+ 1.
Then we must compute the integral

r cp'(up) du.

lu(k)\ U(A)

Inserting the definition of cp' we find

r cp'(up) du

lu(k)\ U(A)

= r r cp (Y(~ ~) (~ ~)~ 'l/J-^1 (y) dy du.

}U(k)\ U(A) }y n,m(k)\ Y n ,m (A) ~

The group U' = U IX Y n,m is the standard unipotent subgroup of GLn associated to

the partition ( n 1 , ... , nr, 1, ... , 1) of n. We may decompose this group in a second

manner. If we let U" be the standard unipotent subgroup of GLn associated to

the partition (n1, ... , nr-1, nr + n - m - 1) of n and let Y" be the subgroup of

GLn obtained by embedding unipotent subgroup of GLnr+n-m-l associated to the

partition (nr, 1, ... 1) into GLn by

y" f---+ (fn1+·

0

+nr-l y~')

then U' = Y" IX U". If we extend the character 'ljJ of Y m,n to U' by making it
trivial on U, then in the decomposition U' = Y" IX U", 'ljJ is dependent only on
the Y" component and there it is the standard character 'ljJ on Y". Hence we may
rearrange the integration to give

r cp'(up) du

lu(k)\ U(A)

( ( ( /1 (J y~') (po OJ)) du^11 '1j;-^1 (y") dy".

= JY^11 (k)\ Y"(A) lu^11 (k)\ U"(A) cp U Q

But since cp is cuspidal on GLn and U" is a standard unipotent subgroup of GLn
then

fu 11 (k)\ U"(A) cp ( u" G y~') (~ n) du"=^0

from which it follows that

r cp'(up) du= 0

lu(k)\ U(A)