LECTURE 5. CONVERSE THEOREMS 155The proof of this Lemma is simply an exercise in the Kirillov model of IIv and
can be found in [12].
If we now fix such a place vo and assume that our vector ~ is chosen so that
~vo = ~~o, then we havefor such~·
Hence we now have Ut,(In) = ~(In) for all~ E Vrr such that ~vo = ~~o at our
fixed place. If we let G' = Koo,vo (p~;o) avo, where we set avo = IJ~#vo GLn(kv),
then we have this group preserves the local component ~~o up to a constant factor
so that for 9 E G' we have Ut,(9) = Urr(g)t,(In) = Vrr(g)f.(In) = Vt,(9).
We now use a fact about generation of congruence type subgroups. Let r 1 =
(P(k) Z(k)) n G', I'2 = (Q(k) Z(k)) n G', and I' = GLn(k) n G'. Then Ut,(9) is
left invariant under I' 1 and Vt, (9) is left invariant under I' 2. It is essentially a
matrix calculation that together r 1 and r 2 generate I'. So, as a function on G',
Ut, (9) = Vt, (9) is left invariant under I'. So if we let rrvo = ®~#vo IIv then the
map ~vo ~ Ut,~ 0 ©t,vo (9) embeds Vrrvo into A(I'\ G'), the space of automorphic
forms on G' relative to r. Now, by weak approximation, GLn(A) = GLn(k) · G'
and I'= GLn(k) n G', so we can extend rrvo to an automorphic representation of
GLn(A). Let II 0 be an irreducible component of the extended representation. Then
II 0 is automorphic and coincides with II at all places except possible v 0.
One now repeats the entire argument using a second place v 1 =f. v 0. Then we
have two automorphic representations II 1 and II 0 of GLn(A) which agree at all
places except possibly Vo and v 1. By the generalized Strong Multiplicity One for
GLn we know that Ila and II 1 are both constituents of the same induced represen-
tation 3 = Ind( 0" 1 0 · · · 0 O"r) where each O"i is a cuspidal representation of some
GLm, (A), each mi 2 1 and I: mi = n. We can write each O"i = O"f 0 I <let It' with
O"i unitary cuspidal and ti E ~ and assume t 1 2 · · · 2 tr. If r > 1, then either
m 1 :::; n - 2 or mr :::; n - 2 (or both). For simplicity assume mr :::; n - 2. Let S be
a finite set of places containing all archimedean places, v 0 , v 1 , Sn, and Sa, for each
i. Taking n' =Cir E T(n - 2), we have the equality of partial L-functions
Ls(s,IIx n') = Ls(s,II 0 x n') = Ls(s,II1 x n')
=II Ls (s, O"i x n') =II Ls (s +ti - tr, O"i x a~).
iNow Ls(s,O"~ x a~) has a pole at s = 1 and all other terms are non-vanishing at
s = l. Hence L(s, II x n') has a pole at s = 1 contradicting the fact that L(s, II x n')
is nice. If m 1 :::; n - 2, then we can make a similar argument using L(s, fix 0" 1 ). So
in fact we must have r = 1 and II 0 = II 1 = 3 is cuspidal. Since II 0 agrees with II
at v 1 and II 1 agrees with II at vo we see that in fact II = Ila = II1 and II is indeed
cuspidal automorphic.