Hecke Equivariance of Divisor Lifting with respect to Sesquiharmonic Maass Forms

TL;DR

This paper proves Hecke equivariance of divisor lifting for sesquiharmonic Maass forms, revealing a Hecke system structure and extending results on twisted traces, advancing understanding of modular forms and their symmetries.

Contribution

It establishes Hecke equivariance of divisor lifting for sesquiharmonic Maass forms and connects this to the structure of related Hecke systems and trace formulas.

Findings

Hecke equivariance of divisor lifting is proven.

Sesquiharmonic Maass functions form a Hecke system similar to $J_n$.

Extension of Matsusaka's results on twisted traces.

Abstract

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.