1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 5. CONVERSE THEOREMS 153

Let v E S. Let Kv = GLn(ov) be the standard maximal compact subgroup of

GLn(kv)· Let Pv C Ov be the unique prime ideal of Ov and for each integer mv 2'. 0

set

Ko,.(P:'") ~ { g E GLn(o^0 )lg o{ ' O ;) (mod pm•)}

and Ki,v(P:;-'v ) = {g E Ko,v(P:;-'v) I 9n,n = 1 (mod p:;-'v))}. Note that for m v = 0

we have Ki,v(Pe) = Ko,v(Pe) = Kv. Then, as noted at the end of Section 3.1.2, for

each local component IIv with v E S there is a unique integer mv 2'. 0 such that the

dimension of the space of K 1 ,v (p:;-'v )-fixed vectors in IIv is exactly one. For every

place v E S we choose a vector ~~ such that ~~ is invariant under the compact open

subgroup K 1 (p:;-'v ) for this value of mv. This vector will necessarily transform by

the character wrrv under the action ofK 0 (p:;-'v).

As is standard, we will let Gs = ITvES GLn(kv), Gs = ITv(tS GLn(kv), IIs =

®vESIIv, IIS = ®~(tsIIv, etc .. Let Ko,s(n) = ITvES Ko,v(P::-'v ) c Gs where n =

ITvES P::-'v. Let ~s = ®vES~~ E Vrr 5. Let ~s be any vector in Vrrs. Then for ~

of the form ~ = ~s ® e the functions UE ( h l) and Ve ( h l) are unramified

at the places v E S, so that the integrals I(s; UE, cp') and J(s; Ve, cp') vanish unless

cp'(h) is also unramified at those places in S. In particular, if n' E T(n - 1) but

n' rf_ ys (n-l) these integrals will vanish for all cp' E V7r'. So now, for this fixed class

of~ we actually have J(s; UE, cp') = J(s; Ve, cp') for all cp' E V7r' for all n' E T(n-1).
Hence, as in Proposition 5.1, UE(In) = Ve(In) for all such ~· Then the previous
argument now lets us conclude that UE(g) = Ve(g) for all g E Ko,s(n) Gs.
Let Po(n) = P(k) n Ko,s(n) Gs, which in fact is simply P(k), and Q 0 (n) =
Q(k) n Ko,s(n) Gs. Then a simple matrix computation shows that Po(n) and
Q 0 (n) generate the congruence type subgroup f 0 (n) = GLn(k) n K 0 ,s(n) Gs of
G' = Ko,s(n) Gs. Hence the mapping e 1--7 UE~®Es(g) embeds Vrrs into the
space of automorphic forms A(r 0 (n)\ G';wrr) as a representation of G'. Since
by approximation GLn(A) = GLn(k)G' and fo(n) = GLn(k) n G' we see that
A(GLn(k)\ GLn(A);wrr) = A(f 0 (n)\ G';wrr) so that IIs determines an automor-
phic representation II 1 of GLn(A). Then by construction, II 1 ,v '.::::'. IIv for all v rf_ S.
For our II' of the theorem we now take any irreducible constituent of IIi.

5.4. Remarks on the proof of Theorem 5.2

Details for this section can be found in [12].

5.4.1. The case of S empty

Now suppose that n 2'. 3, and that L(s, II x n') is nice for all n^1 E T(n - 2). Then
from Proposition 5.1 we may conclude that IP'~ 2 UE(Jn-1) = IP'~ 2 VEUn-1) for all
~ E Vrr. Since the projection operator IP'~ 2 now involves a non-trivial integration
over kn-^1 \An-l we can no longer argue as in the proof of Theorem 5.1. To get to
that point we will have to impose a local condition on the vector ~ at one place.
Before we place our local condition, let us write FE = UE - Ve. Then FE is
rapidly decreasing as a function on Pn-l· We have IP'~ 2 FE(In-1) = 0 and we