Geodesic
On Penney's Cayley Transform of a Homogeneous Siegel Domain
Takaaki Nomura1, 2, 3
1Dept. of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
2We introduce a Cayley transform 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 C, and thus clarify the biholomorphic nature of C in a direct and visible manner. When D is quasisymmetric, our Cayley transform 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.
3[
Journal of Lie Theory, Tome 11 (2001) no. 1, pp. 185-206 
Takaaki Nomura. On Penney's Cayley Transform of a Homogeneous Siegel Domain. Journal of Lie Theory, Tome 11 (2001) no. 1, pp. 185-206. http://geodesic.mathdoc.fr/item/JOLT_2001_11_1_a10/
@article{JOLT_2001_11_1_a10,
     author = {Takaaki Nomura},
     title = {On {Penney's} {Cayley} {Transform} of a {Homogeneous} {Siegel} {Domain}},
     journal = {Journal of Lie Theory},
     pages = {185--206},
     year = {2001},
     volume = {11},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2001_11_1_a10/}
}
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We introduce a Cayley transform 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 C, and thus clarify the biholomorphic nature of C in a direct and visible manner. When D is quasisymmetric, our Cayley transform 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.