Harmonic Weak Siegel Maa{\ss} Forms I

TL;DR

This paper constructs a preimage for non-holomorphic Saito-Kurokawa lifts under a lowering operator, enabling the decomposition of harmonic weak Siegel Maaß forms into mock modular components.

Contribution

It introduces a method to obtain preimages of non-holomorphic lifts, facilitating the study of Siegel mock modular forms.

Findings

Preimages under the lowering operator are constructed.

Harmonic weak Siegel Maaß forms decompose into meromorphic and non-holomorphic parts.

Every harmonic weak Siegel Maaß form corresponds to a Siegel mock modular form.

Abstract

Given a non-holomorphic Saito-Kurokawa lift we construct a preimage under the vector-valued lowering operator. In analogy with the case of harmonic weak elliptic Maa{\ss} forms, this preimage allows for a natural decomposition into a meromorphic and a non-holomorphic part. In this way every harmonic weak Siegel Maa{\ss} form gives rise to a Siegel mock modular form.