Differential Operators for Siegel-Jacobi forms

TL;DR

This paper develops invariant differential operators for Siegel-Jacobi forms by computing the Chern connection on the Siegel-Jacobi space, leading to new Maass-Shimura type operators.

Contribution

It introduces a novel method to derive invariant differential operators for Siegel-Jacobi forms using geometric connections.

Findings

Constructed a series of invariant differential operators.

Derived two types of Maass-Shimura type operators.

Enhanced understanding of differential structures on Siegel-Jacobi space.

Abstract

For any positive integers $n$ and $m$, $\mathbb{H}_{n,m}:=\mathbb{H}_n\times\mathbb{C}^{(m,n)}$ is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. In this article we compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we constructed a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for $\mathbb{H}_{n,m}$ are obtained.