170 J.W. COGDELL, £-FUNCTIONS FOR GLnAs noted above, there is no reason for us to expect the L-functions L(s, n x n')
to be entire for all cuspidal n'. However, one can analyze the potential poles
in terms of the Eisenstein series above. The crucial observation of Kim was the
following [7, 84].
Proposition 6.9. The relevant Eisenstein series, hence the L-function L(s, 7r x n'),
can have poles only if the representation n' is essentially self-contragredient, that
is, ;i ~ n' 0 I det It for some t E C.This condition for poles is a condition on our twisting representation n' and
can again be controlled by a ramified twist. If we assume that n' is such that at
one finite place v we have ?r~ = nb,v 0 T/v with nb,v unramified and T/v a character of
k; such that both T/v and T/; are ramified then we can never have ;, ~ n' 0 I det It
since this is not possible locally at the place v. Hence L(s, n x n') will be entire.
Combining these three results, we have the following statement.
Proposition 6.10. Let 7r be a globally generic cuspidal representation of H(A). Let
S' be a non-empty set of finite places and suppose that T/ is an idele class character
such that at at least one place v E S' we have both T/v and T/; are ramified. Then
the twisted L-functions L(s, n x n') are nice for all n' E T^81 (2n - 1) 0 rJ.6.4.3. Application of the Converse Theorem
We are now ready to complete the proof of Theorem 6.1 in the case of H = S0 2 n+l ·
Let 7r be a globally generic cuspidal representation of H(A). Let S be the finite set
of finite places at which ?rv is ramified. Let IT be the candidate lift of 7r to GL 2 n(A)
constructed above, that is, ITv is the local Langlands lift of 7r v for v tJ. S and ITv is
any irreducible admissible generic representation of GL 2 n(kv) having trivial central
character for v E S. If Sis non-empty let S' = Sand if 7r is unramified at all finite
places take S' = { vo} to contain any chosen finite place. Choose a fixed idele class
character T/ which is sufficiently ramified for all v E S' such that both Propositions
6.6 and 6.10 are valid. Then for all n' E T^81 (2n - 1) 0 T/ we have
L( s, n x n') = L( s, IT x n') and c:( s, n x n') = c:( s, IT x n')
and the L(s, IT x n') are thus nice. Then applying Theorem 5.1 and Observation
5.1 we can conclude that there exists a automorphic representation IT' = @'IT~
of GL2n(A) such that IT~ = ITv is the local Langlands lift of ?rv for a ll v tJ. S'.
Hence IT' is a global Langlands lift of n associated to the embedding of L-groups
Sp 2 n(IC) ,__. GL2n(IC). This is Theorem 6.1 in this case.6.5. Liftings from the other classical groups
The liftings of globally generic cuspidal representations from S0 2 n to GL 2 n and
Sp 2 n to GL2n+1 follows the same outline as above. At the time of writing [7]
the stability of the local 1 -factors was known only for H = S0 2 n+l· Since then,
Shahidi has established formulas for his local coefficients, and hence his local 1 -
factors, which represent them as Mellin transforms of suitable Bessel functions [82].
Having such a representation was crucial for the proof for stability of 1-factors in
the S02n+1 case [11]. Combining the formulas of [82] with the analysis of [11]
now gives the stability in these cases. Having this stability in hand, the proof now
follows the method above. The complete proof in these cases will appear in [8].