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, $\Phi_S$, where $S$ is a complex symplectic matrix, on the Siegel upper half space. We partially classify the complex symplectic matrices for which $\Phi_S(Z)$ is well defined. We also consider $\mathcal S_n$ and $\overline{\mathcal S}_n$ as metric spaces and discuss distance properties of the map $\Phi_S$ from $\mathcal S_n$ to $\mathcal{S}_n$ and $\overline{\mathcal S}_n$ respectively.