Action of complex Symplectic matrices on the Siegel upper half space

The Siegel upper half space, ${\mathcal{S}}_n$, the space of complex symmetric matrices, $Z$ with positive definite imaginary part, is the generalization of the complex upper half plane in higher dimensions. In this paper, we study a generalization of linear fractional transformations, ${\mathrm{\Phi }}_S$, where $S$ is a complex symplectic matrix, on the Siegel upper half space. We partially classify the complex symplectic matrices for which ${\mathrm{\Phi }}_S(Z)$ is well defined. We also consider ${\mathcal{S}}_n$ and ${\bar{\mathcal{S}}}_n$ as metric spaces and discuss distance properties of the map ${\mathrm{\Phi }}_S$ from ${\mathcal{S}}_n$ to ${\mathcal{S}}_n$ and ${\bar{\mathcal{S}}}_n$ respectively.