Action of complex Symplectic matrices on the Siegel upper half space

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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 Sₙ as metric spaces and discuss distance properties of the map Φₛ from Sₙ to Sₙ and Sₙ respectively.