Invariant Differential Operators on Siegel-Jacobi Space

Abstract

For two positive integers $m$ and $n$, we let ${\mathbb{H}}_n$ be the Siegel upper half plane of degree $n$ and let ${\mathbb{C}}^{(m,n)}$ be the set of all $m\times n$ complex matrices. In this article, we study differential operators on the Siegel-Jacobi space ${\mathbb{H}}_n\times {\mathbb{C}}^{(m,n)}$ that are invariant under the natural action of the Jacobi group $Sp(n,\mathbb{R}⋉H_{\mathbb{R}}^{(n,m)}$ on ${\mathbb{H}}_n\times {\mathbb{C}}^{(m,n)}$, where $H_{\mathbb{R}}^{(n,m)}$ denotes the Heisenberg group. We give some explicit invariant differential operators. We present important problems which are natural. We give some partial solutions for these natural problems.

32 pages. arXiv admin note: substantial text overlap with arXiv:arXiv:math/0611389

References in corpus (4)

• Invariant metrics and Laplacians on Siegel-Jacobi space
• Invariant metrics and Laplacians on Siegel-Jacobi Disk
• A note on a fundamental domain for Siegel-Jacobi space
• A Geometrical Theory of Jacobi Forms of Higher Degree

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