Hecke Equivariance of Divisor Lifting with respect to Sesquiharmonic Maass Forms

We investigate the properties of Hecke operator for sesquiharmonic Maass forms. We begin by proving Hecke equivariance of the divisor lifting with respect to sesquiharmonic Mass functions, which maps an integral weight meromorphic modular form to the holomorphic part of the Fourier expansion of a weight 2 sesquiharmonic Maass form. Using this Hecke equivariance, we show that the sesquiharmonic Maass functions, whose images under the hyperbolic Laplace operator are the Faber polynomials $J_n$ of the $j$-function, form a Hecke system analogous to $J_n$. By combining the Hecke equivariance of the divisor lifting with that of the Borcherds isomorphism, we extend Matsusaka’s finding on the twisted traces of sesquiharmonic Maass functions.

💡 Research Summary

The paper studies Hecke operators acting on sesquiharmonic Maass forms of weight 2 and introduces a divisor‑lifting map that sends an integral‑weight meromorphic modular form to the holomorphic part of a weight‑2 sesquiharmonic Maass form. Building on earlier work where the divisor lifting for ordinary meromorphic modular forms is Hecke‑equivariant, the authors define a new family of sesquiharmonic pre‑images (J_{1,n}) of the classical Faber polynomials (J_n). These functions satisfy a logarithmic correction (formula (1.5)) and are used to construct a lifting \