1549380232-Automorphic_Forms_and_Applications__Sarnak_

310 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD

where 'T/v is the character of F* (with cusp form rJ = ®v'T/v on A'F) defining t he
induced representation. Then (2.20) equals

1+frJv(w~)q;;n-ns 1 lxlvd*x

n=l (w;n)-(w;n+l)

00

1 + L 'T/v(w~)(l - q;;l)q;;ns.

n=l

Write rJv(x) = Jxl~v to get

(2.22)

00

1 + 2= q;;nμ;;'" (1 - q;;l)

n=l

1.

1 - q{;v+s

Now, if av is the standard coroot of SL 2 , then

q~v =av (Av),

following our Hr identification. (See the remark below). Here Av E PGL2(C)
represents the semisimple conjugacy class parametrizing 1fv and av is the root of
PGL 2 (C), or the coroot of SL 2. Clearly av(Av) is the eigenvalue for the adjoint
action of Ly on Ln, evaluated at Av. Therefore

(2.23)

We thus get that (2.19) equals

(2.24)

which equals

(2.25)

since m = 1, i.e., r = r 1.

Remark (r or i'?). Let G be as before a split connected reductive F - group. Let

A= Hr: T ___, Hom(X*(T),Z)

be as before. Denote by^0 T = Ker(A) as in [Bl,Car]. Then A(T) = Tf°T. Clearly

(^0) T is the largest compact subgroup of T. Let Xun(T) be the group of characters
of A(T), i.e., unramified characters of T. Then yv is the complex torus for which
X(Tv ) = A(T) = Hom(X(T),Z) = X(T), G being split, which implies A is a
surjection. Consequently Xun(T) = Hom(A(T), <C) = Hom(X(Tv), <C) = yv.
Thus given tv E yv, there exists Xtv E Xun(T) such that
q(A(t),Av) Xtv (t)
(2.26) IAV (t)J,