LECTURE l. FOURIER EXPANSIONS AND MULTIPLICITY ONE 109Therefore, for a given place v the local Kirillov models of any two irreducible
admissible generic smooth unitary representations have a certain P n ( kv )-submodule
in common, namely VTv.
Let us now return to Piatetski-Shapiro's proof of the Strong Multiplicity One
Theorem [66].
Proof: We begin with our cuspidal representations 71" 1 and 71" 2. Since 71" 1 and 7r 2
are irreducible, it suffices to find a cusp form cp E V71" 1 n V71" 2 • Let P~ = P n Zn
be the (n -1, 1) parabolic subgroup of GLn. Then P~(k)\P~(A) is dense in
GLn(k)\ GLn(A). (This follows from the fact that P~ \ GLn ~ ]pm-l and ]pm-^1 (k)
is dense in IP'n-^1 (A).) So it suffices to find jlmf two cusp forms 1Pi E V7r, which agree
on P~ (A). If we let Wi be the central character of 11"i then by assumption w 1 ,v = w 2 ,v
for v r:J. Sand the weak approximation theorem then implies w 1 = w 2. So it suffices
to find two 1Pi which agree on Pn(A). But as in the proof of the Multiplicity One
Theorem, via the Fourier expansion, to show that cp 1 (p) = cp 2 (p) for p E Pn(A) it
suffices to show that W 'Pi (p) = W cp 2 (p). Since we can take each W 'Pi to be of the
form Ilv W'P,,v then this reduces to a question about the local Kirillov models. For
v r:j. S we have by assumption that IC(7r1,v, 1/Jv) = IC(7r2,v, 1/Jv) and for v ES we have
seen that V 7 v C IC(7r1,v, 7/Jv)n/C(7r2,v, 1/Jv)· So we can construct a common Whittaker
function in the restriction of W(7ri,7/J) to Pn(A). This completes the proof. D