1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. EULERIAN INTEGRALS FOR GLn 115

so that cp' is a cuspidal function on Pm+ 1 (A). D

z.From Lecture 1, we know that cuspidal functions on Pm+ 1 (A) have a Fourier

expansion summed over Nm(k)\ GLm(A). Applying this expansion to our projected

cusp form on GLn(A) we are led to the following result.

Lemma 2.2. Let cp be a cusp form on GLn(A). Then for h E GLm(A) we have

the Fourier expansion

JP>~cp(h 1)=fdet(h)f-(n-r;-l) L w'P((6 I~ )(h J_ ))

')'EN,,,(k)\ GL,,,(k) n m n m

with convergence absolute and uniform on compact subsets.

Proof: Once again let

with p E Pm+l(A). Since we have verified that cp'(p) is a cuspidal function on

P m+l (A) we know that it has a Fourier expansion of the form

cp'(p) = L Wcp' ((6 ~) p)

"YEN,,,(k)\ GL,,,(k)

where

Wcp'(P) = { cp'(np)'¢-^1 (n) dn.

J Nrn+l (k)\ Nm+l (A)

To obtain our expansion for JP>:;, cp we need to express the right hand side in terms

of cp rather than cp'.

We have

Wcp'(P) = { cp'(n'p)'¢-^1 (n') dn'

J Nm+1 (k)\ N rn+ 1 (A)

r r cp (y (n'p ~)) '¢-^1 (y) dy '¢-^1 (n') dn'.

= }Nm+1(k)\Nm+1(A) }Yn,m(k)\ Y,,,m(A) Q

It is elementary to see that the maximal unipotent subgroup Nn of GLn can be
factored as Nn = Nm+l ~ Y n,m and if we write n = n'y with n' E Nm+l and
y E Y n,m then 'lfJ(n) = 'lfJ(n')'lfJ(y). Hence the above integral may be written as

Wcpi(p) = L n(k)\Nn(A) cp ( n (~ In_:J) '¢-l(n) dn = Wcp (~ In:_J.

Substituting this expression into the above we find that

JP>~cp (h 1) =I det(h)f-(n-r;-l) L w'P ( (6 In~m) e In-m))
')'ENrn(k)\ GL,,,(k)

and the convergence is absolute and uniform for h in compact subsets of GLm(A).
D