302 FREYDOON SHAHIDI, LANGLANDS-SHAHIDI METHOD
Let H to be any connected reductive algebraic group defined over F. Consider-
ing Has a group over Fv, for each v, we let Hv = H(Fv)· If AF is the ring of adeles
of F , we set H = H(AF ). It may be considered as a restricted product of groups
Hv with respect to H(Ov ) for all v, where H splits over an unramified extension
[Bl]. There will be no restriction if H splits over F. Moreover, in t his case each
Kv = H(Ov ) is a maximal compact subgroup of Gv. We let K = I1v Kv, where
each K v is a good maximal compact subgroup of Hv and K v = H (Ov) for almost
all v(cf. [Bl,Shl]). Then G =PK.
For every algebraic group H over F , let X (H) F denote the group of F rational
characters of H. We let X(H) = X(H)F. Note that if T is a split torus over F,
then X(T)F = X(T). We set
a= Hom(X(M)F,JR) = Hom(X(A)p,JR).
Then a = X(M)F ©z JR= X(A)F ©z JR and a[: = a ©JR C is the complex dual
of a, via (,\ , x © z) = >-(x)z, ,\ E a, x E X(M)F and z E C. For each v, the
embedding X(M)F '--4 X(M)Fv induces a map from av = Hom(X(M)Fv, JR) to a.
There exists a homomorphism HM : M --+ a defined by
exp(x, HM(m)) = IJ lx(mv)lv
vfor every x E X(M)F and m = (mv)· Extend HM to Hp on G by making it t rivial
on N and K.
If pp is half the sum of roots in N , we set a = (pp,Oi.)-^1 pp E a*. It is a
fundamental weight for T (cf. [Shl,Sh2]).
Finally having fixed M with M J T , let e c 6. denote the subset of simple
roots, generating M. We sometimes write Mo for M. Let W be the Weyl group
of T in G. We use WM to denote its Weyl group in M. There exists a unique
element wo E W such that wo(B) c 6. , while w 0 (0i.) < 0. We will always choose
a representative w 0 for w 0 in G(F) and use w 0 to denote each of its components.
We will be more specific about the choice of w 0 later. Finally, let M' be the Levi
subgroup of G generated by w 0 (B). There exists a parabolic subgroup P' :_) B
which has M' as a Levi factor, in fact t he unique one containing T. Let N' be t he
unipotent radical of P' (cf. [Lal,Sh3]).
1.2. £-Groups, £-Functions and Generic Representations
Denote by X(T) = X(T) the character group of T which is the same as X(T)p.
Let X(T) be the group of cocharacters of T , i.e., homomorphisms from Gm =
F* into T = T(F). Let 6. v = 6. v (T) be the set of simple coroots of T , i.e.,
Oi.v: Gm--+ T satisfying Oi.(Oi.v(t)) = t^2. Let
(1.1) 'l/Jo(G) = (X*(T), 6. (T), X*(T), 6. v (T))denote the based root datum. By Chevalley's theorem [Sp], there exist a complex
connected reductive group a v with a maximal torus rv such that
'l/Jo(G)v = (X*(T), 6. v (T), X*(T), 6.(T))
= 'l/Jo(Gv)
= (X*(Tv), 6.(Tv), X*(Tv), 6. v (Tv) ).