Action of complex symplectic matrices on the Siegel upper half space

The Siegel upper half space, Sn, 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, ΦS, where S is a complex symplectic matrix, on the Siegel upper half space. We partially classify the complex symplectic matrices for which ΦS(Z) is well defined. We also consider Sn and S‾n as metric spaces and discuss distance properties of the map ΦS from Sn to Sn and S‾n respectively.