1549380232-Automorphic_Forms_and_Applications__Sarnak_

304 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD

a map

(1.4)

where the product runs over all f3 E D..', sending exp(xf3Xf3) to Xf3· Let 'lj;p b e a

non-trivial character of F. We shall then define a generic character x of U = U(F)

by

(1.5) x(u) = 'lf;p(L,f3xf3),

where ¢(u) = (xf3)f3 and the /3-component of u is exp(xf3X{3)-Conversely, given a

generic character x of U, i.e., one which is non-trivial on every simple root group,

there exists an F -splitting {Xf3}f3 such that xis defined by (1.4) and (1.5).

If 7f is an irreducible (admissible) unitary representation of M = M(F), then

7f is called generic, or more precisely x-generic for a generic character x of u^0 =

(Un M)(F), if there exists a functional A on the space of 7f, called a Whittaker

functional, which is continuous with respect to the seminorm topology defined by

the Hilbert space norm on the space 1i(7r) of 7f if Fis archimedean (cf. [S,Sh4,Sh5])

(continuous with respect to the trivial locally convex topology on H( 7f) for which

every seminorm is continuous if F is non-archimedean) and satisfies

(1.6) (7r(u)x, A) = x(u)(x, A),

u E U^0 , x E 1i(7r) 00 , the subspace of C^00 -vectors. By a theorem of Shalika [SJ,

the space of all the Whittaker functionals on 1i(7r) is at most one-dimensional.

Changing the splitting, we may assume x is defined by (1.4) and (1.5).

Now, assume F is global. Let 'lj; = ®v'l/Jv be a non-trivial character of F\A.p.

Then each 'l/Jv is non-trivial. Moreover, for almost all v, 'l/Jv is unramified, i.e., Ov is

the largest ideal on which 'l/Jv is trivial. The map (1.4) is F - rational and therefore

extends to a map from U = U(Ap) into TI Ga(Ap ), sending U(F) into TI Ga(F).

We then define a character x of U ( F) \ U by ( 1. 5). Consequently, if xv ( Uv) is defined

by (1.5) and 'l/Jv for each v, then x(u) = Ti v Xv(uv), where u = (uv)v EU.

Now, let 7f = ®v1fv is a cuspidal representation of M = M(Ap). Choose a

function cp in the space of 7f and set

(1.7) W"'(m) = { cp(um)x(u)du.

Juo(F)\Uo

We shall say 7f is (globally) x-generic if W"' -/=- 0 for some cp. Then each 1fv is

Xv- generic. If cp = ®v'Pv, 'Pv E 1i(7rv), then

W'P(m) = IJ (7rv(mv)'Pv> Av)

v

(1.8)

v

for some Xv-Whittaker functional Av on 1i(7rv)·

We finally point out that every generic character x of U(F)(U(F)\ U(AF) if F

is global) points to a F-splitting and conversely, each F-splitting defines a generic

character, both by means of (1.4) and (1.5).

Now assume F is local and 1fv is an irreducible admissible representation of

Mv. Let s EC and denote by I(s, 1fv) = I(sa, 1fv), the induced representation

(1.9) J(s,1fv)= Ind 1fv®q~sa,HMv( ))®l.

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