LECTURE 1
Fourier expansions and multiplicity one
In this section we let k denote a global field, A, its ring of adeles, and '1jJ will
denote a continuous additive character of A which is trivial on k. For the basics
on adeles, characters, etc. we refer the reader to Weil [96] or the book of Gelfand,
Graev, and Piatetski-Shapiro [26].
We begin with a cuspidal a utomorphic representation (7r, V1r) of GLn(A). For
us, automorphic forms are assumed to be smooth (of uniform moderate growth)
but not necessarily K 00 - finite at the archimedean places. This is most suitable
for the analytic theory. For simplicity, we assume the central character w-n: of 7f is
unitary. Then V-n: is the space of smooth vectors in a n irreducible unitary represen-
tation of GLn(A). We will always use cuspidal in this sense: the smooth vectors
in an irreducible unitary cuspidal automorphic representation. (Any other smooth
cuspidal representation 7f of GLn(A) is necessarily of the form 7f = 7r^0 ® I <let It
with 7r^0 unitary and t real, so there is really no loss of generality in the unitarity
assumption. It merely provides us with a convenient normalization.) By a cusp
form on GLn(A) we will mean a function lying in a cuspidal representation. By a
cuspidal function we will simply mean a smooth function cp on GLn(k)\ GLn(A) sat-
isfying fu(k)\ U(A) cp(ug) du= 0 for every unipotent radical U of standard parabolic
subgroups of GLn.
The basic references for this section a re the papers of Piatetski-Shapiro [64, 66]
and Shalika [85].
1.1. Fourier ExpansionsLet cp(g) E V-n: be a cusp form in t he space of 7f. For arithmetic applications, and
particularly for the theory of £-functions, we will need the Fourier expansion of
cp(g).
If f ( T) is a holomorphic cusp form on the upper half plane SJ, say with respect
to 812 (Z), then f is invariant under integral translations, f ( T + 1) = f ( T) and thus
has a Fourier expansion of the form
00
J(-r) = L ane2-n:im'.
n=l
101