1549380232-Automorphic_Forms_and_Applications__Sarnak_

308 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD

They both converge for Re( s) > > 0 and have a finite number of simple poles

for Re(s) 2:: 0, none with Re(s) = 0 (cf. [HC,La2,MW1]).

Let I(s, 7r) = ®vf(s, 7rv) be the representation of G = G(AF) induced from

(2.5) 7r 0 exp(sa +pp, HM( )) 0 1.

Let f E V(s, 7r) = 0v V(s, 7rv) be defined by

f(nomoko) = exp(sa +pp, Hp(mo)) ·

(2.6) r (7(ko^1 )T(k )x, X)7r( mok )cpdk,

}KM

with m 0 E M, no E N and k 0 E K. Here x E 1i(T) and x E 1i(f), where f is the

contragredient of T. Moreover

(2.7)

f(nomoko)(e) = (~s(g)x, x)

= <I>s(g),

where the left hand side is the value of the cusp form f(nomoko) at identity and

g = nomoko (cf. [Sh6]).

Observe that

(2.8) E(s,<I> 8 ,g,P) = (E(s,~s,g,P)x, x).

Given f E V(s,7r) and Re(s) >> 0, define the global intertwining operator

A(s, 7r) by

(2.9) A(s,7r)f(g) = { f(w 01 n'g)dn'.

JN,

Finally, if at each v we define a local intertwining operator by

(2.10) A(s, 1rv, wo)fv(g) = { fv(w 01 n' g)dn',

JN~

then

(2.11)

Observe that

(2.12)

and

(2.13)

A(s,7r): I(s,7r)---+ I(-s,wo(7r))

A(s, 1l"v, wo): I(s, 7rv) ---+ I(-s, wo(7rv)).

Using (2.7), we now define

(2.14) (M(s, 7r)<I>s)(g) = A(s, 7r)f(g)(e),

where by the left hand side we understand the value of the cusp form A(s, 7r)f(g)
at e. This is basically the Langlands' M(s, 7r) introduced in [La2], or as denoted
by Harish-Chandra in [HC], his function c(s, 7r).

Constant Term Theorem. The constant term

(2.15) Ep1(s,<I> 8 ,g,P) = { E(s,<I> 8 , n^1 g,P)dn'

jN'(F)\N'

is equal to

(2.16) Ep1(s, 8 ,g,P) = llM,M's(g) + (M(s,7r)s)(g).