106 J.W. COGDELL, £-FUNCTIONS FOR GLn1.2. Whittaker Models and the Multiplicity One Theorem
Consider now the functions Wcp appearing in the Fourier expansion of a cusp form cp.
These are all smooth functions W(g) on GLn(A) which satisfy W(ng) = w(n)W(g)
for n E Nn(A). If we let W(-rr, w) = {Wcp I cp E v7f} then GLn(A) acts on this space
by right translation and the map cp 1---7 Wcp intertwines Vn with W(n,w). W(n,w)
is called the Whittaker model of n.
The notion of a Whittaker model of a representation makes perfect sense over
a local field or even a finite field. Much insight can be gained by pursuing these
ideas over a finite field [ 28 , 67], but that would take us too far afield. So let kv be a
local field (a completion of k for example [26, 96]) and let ( n v, VnJ be an irreducible
admissible smooth representation of GLn(kv)· Fix a non-trivial continuous additive
character Wv of kv. Let W(-iPv) be t he space of all smooth functions W (g) on
GLn(kv) satisfying W(ng) = Wv(n)W(g) for all n E Nn(kv), that is, the space of all
smooth Whittaker functions on GLn(kv) with respect to Wv· This is also the space
of the smooth induced representation Ind~~n(Wv)· GLn(kv) acts on this by right
translation. If we have a non-trivial continuous intertwining V"v ----+ W(wv) we will
denote its image by W(nv, Wv) and call it a Whittaker model of 1fv·
Whittaker models for a representation (nv, VnJ are equivalent to continu-
ous Whittaker functionals on V"v, t hat is, continuous functionals Av satisfying
Av(nv(n)~v) = Wv(n)Av(~v) for all n E Nn(kv)· To obtain a Whittaker functional
from a model, set Av(~v) = W~Je), and to obtain a model from a functional, set
W~Jg) = Av(nv(g)~v)· This is a form of Frobenius reciprocity, which in this con-
text is the isomorphism between HomNn (V"v, C..pJ and HomcLn (V"v , Ind~:n ( wv))
constructed above.
The fundamental theorem on the existence and uniqueness of Whittaker func-
tionals and models is the following.
Theorem 1.2. Let (nv, VnJ b e a smooth irreducible admissible representation of
GLn(kv)· Let Wv be a non-trivial continuous addit ive character of kv. Then the
space of continuous wv-Whittaker functionals on V"v is at most one dimensional.
That is, Whittaker models, if they exist, are unique.This was first proven for non-archimedean fields by Gelfand and Kazhdan [27]
and their results were later extended to archimedean local fields by Shalika [85].
The method of proof is roughly the following. It is enough to show that W(nv) =
Ind~:n ( wv) is multiplicity free, i.e., any irreducible representation of GLn ( kv) oc-
curs in W(wv) with multiplicity at most one. This in turn is a consequ ence of the
commutativity of the endomorphism algebra End(Ind(wv)). Any intertwining map
from Ind( Wv) to itself is given by convolution with a so-called Bessel distribution,
that is, a distribution Bon GLn(kv) satisfying B(n1gn2) = Wv(n1)B(g)wv(n2) for
n1, n2 E Nn(kv)· Such quasi-invariant distributions are analyzed via Bruhat the-
ory. By the Bruhat decomposition for GLn, the double cosets Nn \ GLn / Nn are
parameterized by the monomial matrices. The only double cosets that can support
Bessel distributions are associated to permutation matrices of the form