1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. FOURIER EXPANSIONS AND MULTIPLICITY ONE 105

assumption to conclude that

<p'(p') = L W40^1 ( (7: n p')

1'ENn-2(k)\ GLn-2(k)

L (k)W401(6^1 p^1 ).

o'ENn-1 (k)\ Pn-1

If we substitute this into the expansion for <p(p) we see

<p(p) = L 'P'en-1 (op)

oEPn-1 (k)\ Pn(k)

L 'P8p(l)

oEPn_i(k)\ Pn(k)

W:n' TOp (0^1 ).

oEPn-l (k)\ Pn(k) o'ENn- 1 (k)\ Pn-1 (k)

Now, as before, Nn-1 \ Pn-1 '.:::::'. Nn \Pn-1 and Nn '.:::::'. Nn-1 I>< Y n-1· Thus

w4?~p(6') = 1 'P8p(n'6')1/J-^1 (n') dn'

Nn-1(k)\ Nn-1(A)

and so

= { { <p(yn'6'6p)'ljJ-^1 (Yn-1)1/J-^1 (n') dy dn'

}Nn_i(k)\ Nn_i(A) }y n(k)\ Y n(A)

= r <p(n6'6p)'ljJ-^1 (n) dn

}Nn(k)\Nn(A)

= W 40 (6'6p)

<p(p) =

oEPn-1 (k)\ Pn(k) o'ENn(k)\Pn-1 (k)

L W40(6p)

oENn(k)\ Pn(k)

L w4?((6 ~)p)

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

which was what we wanted.
To obtain the Fourier expansion on GLn from this, if <p is a cusp form on
GLn(A), then for g E n a compact subset the functions <p 9 (p) = <p(pg) form a
compact family of cuspidal functions on Pn(A). So we have

<pg(l) = L w4?g ( (6 n)

1ENn-1(k)\ GLn_i(k)

with convergence absolute and uniform. Hence

<p (g) = L w 4? ( ( 6 ~)^9 )

1ENn_i(k)\ GLn-1(k)

again with absolute convergence, uniform for g E n.