1549380232-Automorphic_Forms_and_Applications__Sarnak_

152 J.W. COGDELL, £-FUNCTIONS FOR GLn

and similarly for (JP'~"\/{, r.p')a· These are both absolutely convergent for all a and

define continuous functions of a on P\Ax. We now have that I(s;U€,r.p') is the

Mellin transform of (JP'~ UE, r.p') a,

I(s;UE,r.p') = { (JP>;~pE,r.p')a lals-1/2 dxa,
JkX\f!!.X
similarly for I(s; VE, r.p'), and that these two Mellin transforms are equal in the
sense of analytic continuation. By Mellin inversion as in Lemma 11 .3.1 of Jacquet-
Langlands [39], we have that (lP'~UE, r.p')a = (JP'~"\/{, r.p')a for all a, and in particular
for a = l. Since this is true for all r.p' in all irreducible subrepresentations of
automorphic forms on GLm(A), then by the weak form of Langlands' spectral
theory for SLm we may conclude that lP'~U( = JP'~ V( as functions on Pm+1(A).
More specifically, we have the following result.

Proposition 5.1. Let II be an irreducible admissible representation of GLn(A) as

above. Suppose that L(s, II x n') is nice for all n' E T(m). Then for each~ E Vrr

we have lP'~U((Im+1) =JP'~ V((Im+1).

All of our Converse Theorems take Proposition 5.1 as their starting point. The
first part of Theorem 5.1 follows almost immediately. In all others we must add
local conditions to compensate for the fact that we do not have the full family of
twists from Theorem 5.1 and then work around them.

5.3. Remarks on the proof of Theorem 5.1

Let us first look at the proof of Theorem 5.1. Details can be found in [9] and [7].

5.3.1. The case of S empty

We now assume that II is as above and that L(s, II x n') is nice for all n' E T(n-1).

Then we have that for all~ E Vrr, lP'~_ 1 UE(In) = lP'~_ 1 V((In)· But form= n-l

the projection operator lP'~_ 1 is nothing more than restriction to P n· Hence we

have U((In) = V((In) for all~ E Vrr. Then for each g E GLn(A), we have U((g) =

Urr(g)E(In) = Vrr(g)((In) = "V{(g). So the map~~ U((g) gives our embedding of

II into the space of automorphic forms on GLn(A), since now UE is left invariant

under P(k), Q(k), and hence all of GLn(k). Since we still have

Nn(k)\P(k)

we can compute that UE is cuspidal along any parabolic subgroup of GLn· Hence
II embeds in the space of cusp forms on GLn(A) as desired.

5.3.2. The case of non-empty S

Now let S be a non-empty set of finite places of k. Since we are only twisting
by representations which are unramified at places in S, we will only be able to
prove t he equality U( (g) = "!{ (g) for a restricted set of ~ and only on a subset of
GLn(A). Since we have not twisted by all of T(n - 1) we are not in a position to
apply Proposition 5.1. To be able to apply that, we will now have to place local
conditions at all v E S.