Differential Operators for Siegel-Jacobi forms

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.