1549380232-Automorphic_Forms_and_Applications__Sarnak_

166 J.W. COGDELL, £-FUNCTIONS FOR GLn

have been able to overcome these difficulties, as will be discussed below, and this

method has been applied in the following cases:

H LH u:LH--fLG LG G

S02n+i Spzn(<C) '--+ GL2n(<C) GL2n

S02n S02n(<C) '--+ GL2n(<C) GL2n

Spzn S02n+1(<C) '--+ GL2n+1(<C) GL2n+1

GL 2 x GL2 GL2(<C) x GL2(<C) ® GL4(<C) GL4

GL2 x GL3 GL2(<C) x GL3(<C) ® GL5(<C) GL5

GL 4 GL4(<C) /\2 GL5(<C) GL5

In this table, the first three maps u : LH --f LG are the natural embeddings, the

next two are the tensor product maps, and the last is the exterior square map.

Theorem 6.1. Let k be a number field. Let H be a split reductive algebraic group

over k from the table above. Let 7r be a globally generic cuspidal representation

of H(A). Then 7r has a global Langlands lift II' to GLN(A) associated to the map

u : LH --f GLN(<C) from the table. More specifically, there is a non-empty finite

set of finite places S and an automorphic representation II' of GLN(A) such that

for all v tJ. S we have II~ is the local Langlands lift of n v with respect to the

L-homomorphism u.

The first case of this Theorem to appear was the tensor product lifting from
GL2 x GL 2 to GL 4 by Ramakrishnan [69]. His method was slightly different from
the one we have outlined here in that he controlled the analytic properties of the
twisted L-functions for H = GL 2 x GL 2 by a combination of both the Langlands-
Shahidi method and integral representations. In addition he used Theorem 5.2 but
made no use of Observation 5.1 or the highly ramified twist. The first case which
was completely treated by the method outlined here was the lifting from S0 2 n+l
to GL2n in [7]. Once this method was understood, particularly the global use of
Observation 5.1, then other liftings could be obtained whenever one could control
the L-functions. The tensor product lifting from GL 2 x GL 3 to GL 6 by Kim and
Shahidi [55] and the exterior square lift from GL 4 to GL 6 by Kim [53] soon followed.
More recently, we have completed the loca l results necessary to complete the liftings
from the other classical groups S0 2 n and Sp 2 n [8]. In addition, the Asai lifting
from GL2 / K to GL 4 /k, where K/k is a quadratic extension , has been analyzed by
Ramakrishnan [70] by a variant of this method and by Krishnamurthy [57] using
the Langlands-Shahidi method to control the L-functions.
We should point out that in a ll cases, particularly those of Kim-Shahidi [55]
and Kim [53], the Theorem we have stated is the starting point of a more complete
analysis of the lifting as well as applications.