1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE l. FOURIER EXPANSIONS AND MULTIPLICITY ONE 103

Since <p is continuous and t he integration is over a compact set this integral is

a bsolutely convergent, uniformly on compact sets. The Fourier expansion takes the

following form.

Theorem 1.1. Let <p E V7r be a cusp form on GLn(A) and Wcp its associated
'lj;-Whittaker function. Then

<p(g)= 2= wcp((-r 1 )9)

7ENn-l (k)\GLn-1 (k)

with convergence a bsolute and uniform on compact subsets.

The proof of this fact is an induction. It utilizes the mirabolic subgroup P n
of GLn which seems to be ubiquitous in the study of a utomorphic forms on GLn.
Abstractly, a mirabolic subgroup of GLn is simply the stabilizer of a non-zero vector
in (either) standard representation of GLn on kn. We denote by Pn the stabilizer
of the row vector en= (0, ... , 0, 1) E kn. So

Pn = {p = e i) lh E GLn-1,y E kn-l} ~ GLn-1 ~ Yn

where

Y n { - Y- (In-^1

Simply by restriction of functions, a cusp form on GLn(A) restricts to a smooth
cuspidal function on Pn(A) which remains left invariant under Pn(k). (A smooth
function <p on Pn(A) which is left invariant under Pn(k) is called cuspidal if

1

<p(up) du= 0

U(k)\ U(A)

for every standard unipotent subgroup UC Pn.) Since Pn ::::> Nn we may define a
Whittaker function attached to a cuspidal function <p on Pn(A) by the same integral
as on GLn(A), namely

Wcp(P) = 1 <p(np)'ljJ-^1 (n) dn.

Nn(k)\ Nn(A)

We will prove by induction that for a cuspidal function <p on P n (A) we have

<p(p)= L wcp(G~ np)
7ENn-1(k)\ GLn_i(k)
with convergence absolute and uniform on compact subsets.
The function on Y n(A) given by y 1---+ <p(yp) is invariant under Y n(k) since
Y n ( k) C P n ( k) and <p is automorphic on P n (A). Hence by standard abeli an Fourier
analysis for Y n ~ kn-l we have as before

<p(p) = <p).. (p)

where

'-P>.(P) = { <p(yp)>--1 (y) dy.

JY,,(k)\ Yn(A)