J. Hilgert, D. Mayer, H. Movasati
Transfer operators for $\Gamma_{0}(n)$ and the Hecke operators for the period functions for $PSL(2,\matbb{Z})$
(414K, Postscript)
ABSTRACT. In this article we report on a surprising relation between the transfer
operators for the congruence
subgroups
$\Gamma_{0}(n)$ and the Hecke operators on the space of period functions for
the modular group
$\PSL (2,\mathbb{Z})$.
For this we study special eigenfunctions of the transfer operators with
eigenvalues $\mp 1$,
which are also solutions of the Lewis equations for the groups $\Gamma_{0}(n)$
and which are determined by eigenfunctions of the transfer operator for the
modular group
$\PSL (2,\mathbb{Z})$. In the language of the Atkin-Lehner theory of old and
new forms one should
hence call them old eigenfunctions or old solutions of Lewis equation.
It turns out that the sum of the components of these old solutions for the
group $\Gamma_{0}(n)$
determine for any $n$ a solution of the Lewis equation for the modular group
and hence also an
eigenfunction of the transfer operator for this group.
Our construction gives in this way linear operators in the space of period
functions for the group
$\PSL (2,\mathbb{Z})$. Indeed these operators are just the Hecke operators
for the period functions
of the modular group derived previously by Zagier and M\"uhlenbruch using the
Eichler-Manin-Shimura
correspondence between period polynomials and modular forms for the modular
group.