Journal of the
Korean Mathematical Society JKMS

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.

Keywords: partial Cayley transform, Siegel--Jacobi space, Siegel--Jacobi disk, Harish--Chandra decomposition, automorphic factors, Jacobi forms

MSC numbers: Primary 32Hxx, 32M10; Secondary 11F50