LECTURE 1. BASIC CONCEPTS 303The L-group LG of G is Le= av x rF, where rF = Gal(F/F). In general, one
carries the action of rF on roots and coroots dually to QV and let LQ = QV ><I rp.
(Observe that cv = Lc^0 .) In fact, we have an exact sequence
(1.2) 0---+ Int(G)---+ Aut(G)---+ Aut'ljl 0 (G)---+ 0,
where Int(G) is the subgroup of inner automorphisms. One can show that if
{X,e},eEti.', D..' being the set of simple roots of T, is a set of simple root vectors
invariant under r p, then
Aut 'ljlo(G) = Aut(G, B, T, {X,e},eEti.' ).
The set {X,e},eEti.' is ca lled a splitting, as it splits (1.2). The map
rF--+ Aut(G, B , T, {X,e},e)
defines a map
rF--+ Aut'ljlo(G)v = Aut 'l/Jo(Gv )
which is a subset of Aut(Gv). This defines the action of rF on av and defines Le
(cf. [Bl, Sat]).
Given a connected reductive algebraic group Hover F, let L H be its L- group.
Considering Has a group over Fv, we then denote by L Hv its L-group over Fv. If
G is split over F and if we decide to only consider Leo = av, then we may assume
that the L-groups are all the same, no matter the place v. Finally, the natural map
rFv ---+ rF leads to a map 'T/v: L Mv---+ L M for all v.
In our setting L M is a Levi subgroup of LG and one can define a unipotent
group L N (cf. [Bl]) so that L ML N is a parabolic subgroup of LG with unipotent
radical L N. The L-group L M acts on the (complex) Lie algebra Ln of L N. Let T
m
be this representation. Decompose T = EB Ti to its irreducible subrepresentations,
i=l
indexed according to the values (a, (3) = i as (3 ranges among the positive roots of
T. More precisely, X,ev E Ln lies in the space Vi of Ti if and only if (a, (3) = i.
Here X ,ev is a root vector attached to the coroots (Jv, considered as a root of the
L-group. Clearly the integer m is equal to the nilpotence class of Ln. We let
Ti,v =Ti · 'T/v for each i. Again if G is split over F, we may assume Ti,v =Ti for all
v and i (cf. [La,Shl,Sh2]).
Let 7r = ©v1rv be a cuspidal representation of M = M(Ap). Then for almost
all v, 1rv is an unramified representation of M v = M(Fv )· This means that 1rv
has a vector which remains invariant under M( Ov). In this case, the class of 7r v
is determined by a semisimple L M-conjugacy class Av C L Mv = L M. Given a
complex analytic (finite dimensional) representation p of L M, we define the local
Langlands L-function attached to 7rv, panda complex numbers by
(1.3)When Mis quasisplit over Fv, to split over an unramified extension Lw/ Fv, and Tv
is the unique Frobenius conjugacy class in Gal(Lw/ Fv ), then A v must be replaced
by tv ><1 Tv, tv E Lyo. We may moreover assume that tv is fixed by Tv ([Bl, Lal,
Sh2]).
In these lectures, we will mainly be concerned with the case where p = Ti,v, i =
l, ... ,m.
We shall now discuss the notion of generic representations. We will first assume
that Fis a local field. Fix a F -splitting {X,e},eEti.' as before. This then determines