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The standard realization of the Hecke algebra on classical holomorphic cusp forms and the corresponding period polynomials is well known. In this article we consider a nonstandard realization of the Hecke algebra on Maass cusp forms for the Hecke congruence subgroups Γ 0 ( n ). We show that the vector valued period functions derived recently by Hilgert, Mayer and Movasati as special eigenfunctions of the transfer operator for Γ 0 ( n ) are indeed related to the Maass cusp forms for these groups. This leads also to a simple interpretation of the “Hecke like” operators of these authors in terms of the aforementioned nonstandard realization of the Hecke algebra on the space of vector valued period functions.

A realization of the Hecke algebra on the space of period functions for Γ 0 ( n )