Journal of Lie Theory, Vol. 11, No. 1, pp. 185-206 (2001)

On Penney's Cayley transform of a homogeneous Siegel domain

Takaaki Nomura

Department of Mathematics
Faculty of Science
Kyoto University
Kyoto 606-8502

Abstract: In this paper we introduce a Cayley transform $\cal{C}$ of a homogeneous Siegel domain $D$ as a slight modification of Penney's one. We give an explicit formula to the inverse map of $\cal{C}$, and thus clarify the biholomorphic nature of $\cal{C}$ in a direct and visible manner. When $D$ is quasisymmetric, our Cayley transform $\cal{C}$ is shown to be naturally coincident with Dorfmeister's one. A phenomenon which does not appear in the case of quasisymmetric domains is presented by an example in the last section.

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