A Partial Cayley Transform of Siegel-Jacobi Disk

Abstract

Let ${\mathbb{H}}_g$ and ${\mathbb{D}}_g$ be the Siegel upper half plane and the generalized unit disk of degree $g$ respectively. Let ${\mathbb{C}}^{(h,g)}$ be the Euclidean space of all $h\times g$ complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk ${\mathbb{D}}_g\times {\mathbb{C}}^{(h,g)}$ onto the Siegel-Jacobi space ${\mathbb{H}}_g\times {\mathbb{C}}^{(h,g)}$ which gives a partial bounded realization of ${\mathbb{H}}_g\times {\mathbb{C}}^{(h,g)}$ by ${\mathbb{D}}_g\times {\mathbb{C}}^{(h,g)}$. We prove that the natural actions of the Jacobi group on ${\mathbb{D}}_g\times {\mathbb{C}}^{(h,g)}$ and ${\mathbb{H}}_g\times {\mathbb{C}}^{(h,g)}$ are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differential operators on the Siegel-Jacobi disk ${\mathbb{D}}_g\times {\mathbb{C}}^{(h,g)}$ invariant under the natural action of the Jacobi group on ${\mathbb{D}}_g\times {\mathbb{C}}^{(h,g)}$ explicitly.

12 pages Added references

References in corpus (2)

• A note on a fundamental domain for Siegel-Jacobi space
• A Geometrical Theory of Jacobi Forms of Higher Degree