Lecture 6. Converse theorems and functoriality
In this section we would like to make some general remarks on how to apply
these Converse Theorems to the problem of functorial liftings [3]. Other surveys of
this topic can be found in [14, 83].
In order to apply these these theorems, you must be able to control the global
properties of the L-function. However, for the most part, the way we have of con-
trolling global L-functions is to associate them to automorphic forms or represen-
tations. A minute's thought will then lead one to the conclusion that the primary
application of these results will be to the lifting of automorphic representations
from some group H to GLn. This has traditionally been the case, for example in
Shimura's original proof of the Shimura correspondence [86], the Doi-Naganuma
analysis of quadratic base change for GL2 [18], and the symmetric square lifting
from GL 2 to GL 3 by Gelbart and Jacquet [23]. More explicitly number-theoretic
applications then come as consequences of these liftings.
In the recent cases in which the Converse Theorem has been used to establish
Functorial liftings, the group H has been split and the field k has been of charac-
teristic zero. To simplify our exposition we will work in this context throughout
this lecture. So let k be a number field and H a split connected reductive algebraic
group over k.
6.1. Functoriality
Langlands' Principle of Functoriality is a natural philosophy governing the lifting or
transfer of automorphic representations, having its origins in viewing the Langlands
Conjectures as giving an arithmetic parameterization of local admissible or global
automorphic representations.
6.1.1. Langlands Conjectures
Let LH be the Langlands L-group of H. Since we are assuming that His split, the
Galois structure will play no role and we can simply use the connected component
LHO as the full L-group without loss of information. This connected component
is essentially the complex analytic group determined by the root data which is
dual to that of H [3, 6]. The Langlands Conjectures can be viewed as giving an
arithmetic parameterization of either the admissible representations of H( kv) or the
automorphic representations of H(A) in terms of admissible homomorphisms of the
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