LECTURE 1. FOURIER EXPANSIONS AND MULTIPLICITY ONE 107and the resulting distributions are then stable under the involution g r--; gcr =
Wn 'g Wn with Wn ~ C · ·
1
) the long Weyl element of GLn. Thudodhe
convolution of Bessel distributions we have B 1 B 2 = (B 1 B 2 )cr = BfJ. Bf = B 2 B 1.
Hence the algebra of intertwining Bessel distributions is commutative and hence
W( Viv) is multiplicity free.
A smooth irreducible admissible representation (7rv, V7rJ of GLn(kv) which pos-
sesses a Whittaker model is called generic or non-degenerate. Gelfand and Kazhdan
in addition show that 1fv is generic iff its contragredient ifv is generic, in fact that
if '.:::'. ?r" where i is the outer automorphism g" = tg-^1 , and in this case the Whit-
taker model for ifv can be obtained as W(irv,?/i;;-^1 ) = {W(g) = W(wn tg-^1 ) I WE
W(7r, Viv)}.
As a consequence of the local uniqueness of the Whittaker model we can con-
clude a global uniqueness. If ( 7r, V1r) is an irreducible smooth admissible representa-
tion of GLn(A) then 7r factors as a restricted tensor product of local representations
7r '.:::'. rg/1fv taken over all places v of k [19, 26]. Consequently we have a continuous
embedding V1rv '---> V1r for each local component. Hence any Whittaker functional
A on V1r determines a family of local Whittaker functionals Av on each V1rv and
conversely such that A = ®'Av. Hence global uniqueness follows from the local
uniqueness. Moreover, once we fix the isomorphism of V1r with ®'V1rv and define
global and local Whittaker functions via A and the corresponding family Av we
have a factorization of global Whittaker functions
W~(g) =II W~Jgv)
vfor ~ E V7r which are factorizable in the sense that ~ = ®'~v corresponds to a
pure tensor. As we will see, this factorization, which is a direct consequence of the
uniqueness of the Whittaker model, plays a most important role in the development
of Eulerian integrals for GLn.
Now let us see what this means for our cuspidal representations (7r, V7r) of
GLn(A). We have seen that for any smooth cusp form cp E V7r we have the Fourier
expansion
cp(g)= 2: w~((-r 1)g).
')'ENn-1 (k)\GL,,-1 (k)
We can thus conclude that W(7r, 1/1) =f. 0 and that 7r is (globally) generic with
Whittaker functional
A(cp) = W~(e) = J cp(ng)1ji-^1 (n) dn.
Thus cp is completely determined by its associated Whittaker function W~. From
the uniqueness of the global Whittaker model we can derive the Multiplicity One
Theorem of Piatetski-Shapiro [66] and Shalika [85].
Theorem (Multiplicity One). Let (7r, V7r) be an irreducible smooth admissible
representation of GLn(A). Then the multiplicity of 7r in the space of cusp forms on
GLn(A) is at most one.
Proof: Suppose that 7f has two realizations ( 7f 1 , V7ri) and ( 7r2, V7r 2 ) in the space of
cusp forms on GLn(A). Let 'Pi E V7r, be the two cusp forms associated to the vector