1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 215

where

o(it k) _ r(1/2 +it - k/2)

' - r(1/2 +it+ k/2)'

then Qit,k is null if it is of the form it = k':z^1 for some k' = k(2), k' ~ k, and

defines an isometric involution on Sk(q, x, it) if it is not. In the latter case, if f is an

eigenfunction of Qit,k with eigenvalue 61 = ±1, one has the following symmetry:

f(l/2 +it+ k/2)

P1(-n) = 6tr(l/2 +it - k/2)Pt(n).

2.2. Hecke operators

For n ?;: 1 the Hecke operator Tn is defined on automorphic functions of weight k

by

(Tnf)(z) = fa^1 '°' L.,, x(a) '°' L.,, f(-d-). az + b

ad=n b(d)

These operators form a commutative algebra: more precisely, one has the relation

TmTn = L x(d)Tmn/d2·
dl(m,n)
Since Tn commutes with t::..k, Tn acts on each eigenspace Sk(q, x, it) and on the
Eisenstein subspace. Moreover the Tn with (n, q) = 1 are normal on Lk(q, x); more
precisely one has for any f, g,
(Tnf, g) = x(n)(f, Tng),
and in particular, one can choose an orthonormal basis composed of common eigen-
values of t::..k and of the Tn for (n, q) = 1. Such a distinguished basis will be called
a Hecke eigenbasis. For f a Hecke cusp form, we denote by >..1 ( n) its n-th Hecke
eigenvalue; one has

>..1(n) = x(n)>..1(n), >.. 1 (m)>.. 1 (n) = I: x(d)>.. 1 (mn/d^2 ),

dl(m,n)

and by Mobius inversion

>..1(mn) = L μ(d)x(d)>..1(m/d)>..1(n/d)

dl(m,n)

for (mn, q) = 1. Note that the two relations above hold for all m, n if f is an
eigenvalue of all the Hecke operators. Applying the Hecke operators to the Fourier
expansion (2.7), one finds also the simple proportionality relations: for n?;: 1,
(2.10) P1(n) = P1(l)>..1(n)n-^1 l^2 ,p1(-n) = P1(-l)>..1(n)n-^1!^2.
By Atkin/Lehner/Li theory, the whole Hecke algebra can be diagonalized fur-
ther inside the subspaces of new-forms Sk(q, x, it) say. Recall that by definition
sr(q, x, it) is the orthogonal complement inside Sk(q, x, it) of the subspace gener-
ated by the forms
f(dz), f E Sk(q'q,x',it),dq'lq/q, q'q* < q

where q is the conductor of the primitive character x underlying x and x' is
its induced character. We recall as well that the strong multiplicity one property
holds in Sk(q, x, it). This means that a new Hecke eigenform (i.e. belonging to