LECTURE 6. CONVERSE THEOREMS AND FUNCTORIALITY 165an appropriate local lift 7r v f---7 IIv associating to 7r v an irreducible admissible rep-
resentation of GLN(kv)· If the local Langlands conjecture is known for H(kv) we
take IIv to be the local Langlands lift of 1rv as defined above. At the remaining
places, if any, the existence of a local lift is a problem that will be addressed be-
low. Putting these local lifts together we obtain a candidate lift II = @'IIv which
is an irreducible admissible representation of GLN(A) as in the statement of the
Converse Theorems. One must take care that for each 7r^1 E T for an appropriate
twisting set T we have equalities
L(s, 7r x 7r^1 , u 0 l) = L(s, II x 7r^1 )
and
c(s, 7r x 7r^1 , u 0 l) = c(s, II x 7r^1 ).- Control the analytic properties of the twisted £-functions for H. In our
examples this will be done using the Langlands-Shahidi method as explained in
Shahidi's article in this volume [84]. To apply the Langlands- Shahidi method
we must at present assume that k is a number field, as we have, and that the
cuspidal representation 7r is globally generic [84]. Then we need to know that for
an appropriate twisting set T the twisted £-functions L(s, 7r x 7r^1 , u 0 l) are nice,
ie, for 7r^1 in an appropriate twisting set T we need
(1) L(s,7r x 7r^1 ,u 0 l) and L(s,if x ;(t,u 0 l) have analytic continuations to
entire functions of s,
(2) these entire continuations are bounded in vertical strips of finite width,
nd
(3) they satisfy the standard functional equation
L(s, 7r x 7r^1 , u 0 l) = c(s, 7r x 7r^1 , u 0 l)L(l - s , if x :;;,, u 0 l).
The functional equation (3) is known in wide generality [80, 84]. The bound-
edness in vertical strips (2) is likewise known [25, 84]. After a moments thought
one realizes that the entirety (1) will not be true in general, since certain cuspidal
7r of H(A) are expected to lift to non-cuspidal II on GLN(A) and hence the twisted
£-functions L(s, II x 7r^1 ) need not be entire. This is a difficulty that we will also
address below.
- Application of the appropriate Converse Theorem. In all the examples in
which we have been able to carry out this program, the Converse Theorem that is
used is either Theorem 5.1 or 5.2 in conjunction with Observation 5.1 of Section 5.6.
The use of Observation 5.1 with 'T/ a sufficiently highly ramified idele class character
will be used to solve the local and global difficulties remaining in steps 1 and 2.
Once we apply the Converse Theorem we can conclude that there is a automorphic
representation II' = @'II~ of GLN(A) such that II~ = IIv is the local Langlands lift
of 1rv for all v outside a finite set of places S, i.e, II' is a global Langlands lift of 7r
with respect to the £-homomorphism u. Thus the global Langlands Functoriality
from H to GLn associated to the £-homomorphism u is established.
6.3. Statement of Results
We left two problems open in the sketch above: (i) the lack of knowledge of the
Local Langlands Conjecture at certain places for H(kv), and hence the lack of a
natural local lift, and (ii) the possibility of global poles for L(s,7r x 7r^1 , u 0 l). We