1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 6. CONVERSE THEOREMS AND FUNCTORIALITY 167

6.4. Example: The lifting from S02n+1 to GL2n

In this section we would like to give a more detailed sketch of the proof of The-

orem 6.1 in the case of the lifting from S02n+l to GL 2 n associated to the em-

bedding of L-groups Sp 2 n(C) ~ GL2n(C). Since the L-functions for S0 2 n+l as-

sociated to this representation of LH are t he standard L-functions, we will omit

t he L-homomorphism from our notation for the L-functions and c:-factors. T he

L-functions and c:-factors for H below are t hose defined by the Langlands-Shahidi

method [84]. More details can be found in [7].

Recall that k is taken to be a number field. For definiteness, we will take H to

be the 'plit 'pecial o'thogonal grnup with 'espect to the Imm (

1

· ·

1

). Let

7r = Ci5/1fv be a globally generic cuspidal representation of H(A).

6.4.1. Construction of a candidate lift

Let S be the finite set of finite places at which t he local component 7r v of 7r is
ramified.
For v t/: S the Local La nglands Conjecture is known for H(kv) and we can
associate to 7r v its local Langlands lift Ilv from the local lift ing diagram:

7f v f------7 1------7 Ilv.

In these cases we have the following proposition, as is expected from t he for-
malism.

Proposition 6.1. Let v tf: S and let Ilv be t he lo cal Langlands lift of 1fv as above.
Let 7r~ be an irreducible admissible generic representation of GLm(kv) with m < 2n.
Then

L(s, 1fv X 7r~) = L(s, Ilv X 7r~) and c:(s, 1fv X 7f~, 7/Jv) = c:(s, Ilv X 7f~, 7/Jv)·
Now we come to the places v E S where we do not have t he Local Langlands
Conjecture at our disposal. Instead, we will replace it with the following two local
facts about representations of H(kv)· As was the case for linear groups, there is a
local 1-factor 1(s, 1fv x 7r~, 7/Jv) for representations of H(kv), where 1fv is our generic
representation of H(kv) and 7r~ is a generic representation of GLm(kv) [80,84]. It
is related to t he local L-and c:-factors by

(^1

) _ E(S,1fv X 1f~,7/Jv)L(l - s,ifv X if~)

I S,1fv x 1fv,7/Jv - L( ').

S,1fv X 1fv

The following two properties of the local 1 - factor are crucial to our local lifting. The
first is the mutliplicativity of local r-factors and is known in quite some generality
[81,84].