1549380232-Automorphic_Forms_and_Applications__Sarnak_

108 J.W. COGDELL, £-FUNCTIONS FOR GLn

~ E V7f. Then we have two nonzero Whittaker functionals on V7f, namely Ai(~) =

W'P, ( e). By the uniqueness of Whittaker models , there is a nonzero constant c such

that A 1 = cA2. But then we have W'P 1 (g) = A1(n(g)O = cA2(n(g)O = cW'P 2 (g)

for all g E GLn(A). Thus

cp1(g) = 2= w'Pl ( (' 1) g)

1ENn-l (k)\GLn- 1 (k)

= c L W <p 2 ( (^1 1 ) g) = ccp2 (g) ·

1ENn-l (k)\GLn-1 (k)

But then V7f 1 and V7f 2 have a non-trivial intersection. Since they are irreducible

representations, they must then coincide. D

1.3. Kirillov models and the Strong Multiplicity One Theorem

The Multipli city One Theorem can be significantly strengthened. The Strong Mul-

tiplicity One Theorem is the following.

Theorem (Strong Multiplicity One). Let (n 1 , V7fJ and (n2, V7f 2 ) be two cuspidal

representations of GLn(A). Suppose there is a finite set of places S such that for

all v eJ. S we have 7r1,v '.:::::'. 7r2,v· Then 7r1 = 7r2.

There are two proofs of this theorem. One is based on the theory of £-functions

and we will come to it in Lecture 4. The original proof of Piatetski-Shapiro [66] is

based on the Kirillov model of a local generic representation.

Let kv be a local field and let (nv, V7fJ be a n irreducible admissible smooth

generic representation of GLn (k v ), such as a local component of a cuspidal rep-

resentation ?r. Since ?rv is generic it has its Whittaker model W(nv, 'l/Jv)· Each

Whittaker function W E W(nv, 'l/Jv), since it is a function on GLn(kv), can be

restricted to the mirabolic subgroup Pn(kv). A fundamental result of Bernstein

and Zelevinsky in the non-archimedean case [1] and Jacquet and Shalika in the

archimedean case [45] says that the map ~v f-) W~vlPn(kv) is injective. Hence the

representation has a realization on a space of functions on P n ( kv). This is the

Kirillov model

K(nv, 'l/Jv) = {W(p)IW E W(nv, 'l/Jv)}.

P n ( kv ) acts naturally by right translation on IC ( 7r v, 'l/Jv) and the action of all of

GLn(kv ) can be obtained by transport of structure. But for now, it is the structure

of !C(n v, 'l/Jv) as a representation of P n(kv) which is of interest.

For kv a non-archimedean field, let (Tv, V 7 J be the compactly induced repre-

sentation Tv = ind~:~~:~ ( 'l/Jv). Then Bernstein and Zelevinsky have analyzed the
representations of P 11 (k v) and shown that whenever ?rv is an irreducible admis-
sible generic representation of GLn(kv) then !C(nv,'l/Jv) contains V 7 v as a Pn(kv )
sub-representation and if ?rv is supercuspidal then !C(nv, 'l/Jv) = Vrv [l].
For kv archimedean, then we let ( Tv, V 7 J be the smooth vectors in the irre-
ducible smooth unitarily induced representation lnd~:~~:~('l/Jv)· Then Jacquet and
Shalika have shown that as long as ?rv is an irreducible admissible smooth uni-
tary representation of GLn(kv) then in fact !C(nv, 'l/Jv) = Vrv as representations of
Pn(kv ) [45, Remark 3.15].