The unique basis functions $j_m$ of the form $q^{-m}+O(q)$ for the space of weakly holomorphic modular functions on the full modular group form a Hecke system. This feature was a critical ingredient in proofs of arithmetic properties of Fourier coefficients of modular functions and denominator formula for the Monster Lie algebra.

In this talk, we consider the basis functions of the space of harmonic weak Maass functions of an arbitrary level, which generalize $j_m$, and show that they form a Hecke system as well. As applications, we establish some divisibility properties of Fourier coefficients of weakly holomorphic modular forms on modular curves of genus $\ge1$. Furthermore, we present a general duality relation that these modular formssatisfy.
This is a joint work with Daeyeol Jeon and Soon-Yi Kang.