The skew-Maass lift I

TL;DR

This paper introduces a new explicit construction of a non-holomorphic Maass lift from harmonic Maass-Jacobi forms to scalar-valued Maass-Siegel forms, expanding the classical holomorphic Maass lift framework.

Contribution

It provides the first explicit, linear construction of a non-holomorphic Maass lift applicable to non-eigenforms, using new techniques for Fourier series analysis.

Findings

Constructed a Maass lift from harmonic Maass-Jacobi forms to Maass-Siegel forms.

Developed new methods for Fourier series expansions of Siegel modular forms.

Established a foundation for further non-holomorphic Maass lift studies.

Abstract

The classical Maass lift is a map from holomorphic Jacobi forms to holomorphic scalar-valued Siegel modular forms. Automorphic representation theory predicts a non-holomorphic and vector-valued analogue for Hecke eigenforms. This paper is the first part of a series of papers. In this series of papers, we provide an explicit construction of the non-holomorphic Maass lift that is linear and also applies to non-eigenforms. In this first part, we develop new techniques to study Fourier series expansions of Siegel modular forms, which allow us to construct a Maass lift from harmonic Maass-Jacobi forms to scalar-valued Maass-Siegel forms.