162 J .W. COGDELL, £-FUNCTIONS FOR GLn
lo cal Weil-Deligne group Wt or the conjectural global Langlands group L k into
the Langlands L-group LH.
We will begin with the Local Langlands Conjecture. So let v be a place of k, kv
the corresponding completion, and W~ the associated Weil-Deligne group [92]. If
kv is archimedean, we simply take Wt ~ Wku to be the Weil group. Following Borel
[3, 6] we let <I?(Hv) denote the set of admissible homomorphisms ¢v : Wt --+ LH
modulo inner automorphisms. In the case H = GLn these are simply the Frobenius
semi-simple complex representations of the Weil-Deligne group and for other H
t hey are an appropriate generalization. Let A(Hv) = A(H(kv)) denote the set of
equivalence classes of irreducible admissible complex representations of H(kv)·
Local Langlands Conjecture: There is a surjective map A(Hv) --+ <I?(Hv) with fi-
nite fibres which partitions A(Hv) into disjoint finite sets A¢v = A
certain representation-theoretic desiderata.
For the precise nature of the desiderata we refer the reader to Borel [3] or [6].
Since they will play no role in our discussion we will refrain from listing them. The
sets A ¢v for ¢v E <I?(Hv) are called local £-packets.
The following general results are known towards this Conjecture for H:
- If H = GLn then the Local Langlands Conjecture for GLn in characteristic
zero h as been completely established by Harris-Taylor [32] and then Henniart [35].
In this case the correspondence is bijective and the desiderata are expressed in terms
of the matching of £-factors and €-factors of pairs. - If the local field kv is archimedean, i.e., kv =JR or <C, then it was completely
established by Langlands [62]. - If kv is non-archimedean one knows how to parameterize the unramified
representations of H(kv) via the unramified admissible homomorphisms [3]. This
is a rephrasing in this langu age of the Satake classification. - If kv is non-archimedean then Kazhdan and Lusztig have shown how to
parameterize those representations of H(kv) having an Iwahori fixed vector in terms
of admissible homomorphisms of the Weil-Deligne group [50].
Thinking of the Local Langlands Conjecture as providing an arithmetic para-
meterization of the irreducible admissible representations of H(kv), one can define
local £ -functions associated to arbitrary 7rv E A(Hv)· One nee ds a second parame-
ter, namely a continuous complex representation r: LH--+ GLn(<C). Then , for any
admissible homomorphism ¢v E <I?(Hv), the composition ro¢v : Wt --+ GLn(<C) is a
continuous complex representation of the Weil-Deligne group and to it we can asso-
ciate an £-factor L(s, ro¢v) and €-factor E(s, ro¢v, 'l/Jv) for an additive character 'l/Jv
of k [92]. If 7rv E A
¢v then we set
L(s, 7T"v, r) = L(s, r o ¢v) and
According to this definition, one cannot distinguish between the representations 7r v
lying in a given £-packet A¢v in terms of their £-functions and €-factors, hence
the terminology. At present these £-functions are well-defined only for those 7rv for
which the parameterization is known, for example if 7rv is unramified.
One would ideally like a statement of a Global Langlands Conjecture or para-
meterization which is analogous to the local one, but at present there is no natural
global version of the Weil-Deligne group in characteristic zero. One can give such
a formulation in terms of a conjectural global La nglands group Lk for a number