LECTURE 6. CONVERSE THEOREMS AND FUNCTORIALITY 169Proposition 6.5. Let v E S and let IIv be the local lift of 7rv as above, that is,
any generic irreducible admissible representation of GL 2 n(kv) having trivial central
character. Let 7r~ be an irreducible admissible generic representation of GLm(kv)
with m < 2n of the form 7r~ = 7rb,v 0 T/v with 7rb,v unramified and T/v a fixed
sufficiently highly ramified character of k;:. Then
L(s, 1rv X 7r~) = L(s, IIv X 7r~) and c(s, 1rv X 7r~, 'l/;v) = c(s, IIv x 7r~, 'l/;v)·
We will sketch the proof of this Proposition on the level of 1-factors. Since
7rb,v is generic and unramified, it is a full induced representation of the form 7rb,v =
Ind(μ1,v©· · · © μm,v) with each μi,v(x) = lxl~i an unramified character of k;:. Then
7r~ = 7rb,v 0 T/v = Ind(μi,vT/v 0 · · · 0 μm,vT/v) and we have
')'(S, 1rv X 7r^1 V> ¢v) = 1(s, 1rv X Ind(μi,vT/v © · · · © μm,vT/v), 'l/;v)
=II ')'(s +Si, 1rv X T/v, ¢v) (multiplicativity)= II 1( s + Si, IIv X T/v, 'l/;v) (comparing stable forms)
= 1(s, II v X Ind(μi,vT/v 0 · · · 0 μm,vT/v), ¢v) (multiplicativity)
= 1(s, IIv X 7r~, 'l/;v)·
L,From this one derives the equality for the L-and t:-factors.
We can now construct our candidate lift. With 7r and S as above we take IIv
to be the local Langlands lift of 7rv for all v ~ S and take IIv to be any irreducible
admissible generic representation of G L 2 n ( kv) with trivial central character for
v E S. Let II = ©'IIv. Then II is an irreducible admissible representation of
GL 2 n(A). From Proposition 6.1 and 6.5 we may now deduce the following result.
Proposition 6.6. Let 7r and S be as above and let II be the candidate lift of
7r constructed above. Then for any fixed idele class character T/ for which T/v is
sufficiently ramified at the places v E S so that Proposition 6.4 holds we have
L(s, 7r x 7r^1 ) = L(s, II x 7r^1 ) and c(s, 7r x 7r^1 ) = c(s, II x 7r^1 ).
for all 7r^1 E T^8 (2n - 1) 0 T/·
6.4.2. Controlling the analytic properties of the twisted L-functions
The twisted L-functions L(s, 7r x 7r^1 ) are controlled using the Langlands-Shahidi
method. We refer to Shahidi's article in these proceedings [84] for a discussion of
this method. In our case, these results are obtained by analyzing the Eisenstein
series on S0 2 (n+m)+l induced from the representation ;'I det Is ©7r on the maximal
parabolic with Levi subgroup GLm x S0 2 n+l as well as the Eisenstein series on
S0 2 m+l induced from the representation 7r^11 <let ls/^2 on the maximal parabolic with
Levi subgroup GLm for all m = 1, ... , 2n - 1.
The general functional equation has been understood for many years [80, 84]:
Proposition 6.7. For any cuspidal representation 7r^1 of GLm(A), 1 ~ m < 2n, we
have the functional equation
L(s, 7r x 7r^1 ) = E(s, 7r x 7r^1 )L(l - s, ii" x ir').
Similarly, the boundedness in vertical strips is true for all 7r^1 [25, 84]:Proposition 6.8. For any cuspidal representation 7r^1 of GLm(A), 1 ~ m < 2n, the
L-function L(s, 7r x 7r^1 ) is bounded in vertical strips.