Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms
In the present paper, the author's main purpose is to construct a family of Cayley transforms which carry a homogeneous Siegel domain \(D\) birationally and biholomorphically onto a bounded domain in complex \(n\)-space. In doing so, the author gathers under one umbrella a variety of previously-studied Cayley transforms which, while not essentially different in the case that \(D\) is symmetric and irreducible, are different in case \(D\) is a non-quasisymmetric Siegel domain of type II. To construct the family of transforms, let \(G\) be a split solvable Lie group acting transitively on \(D\) with Lie algebra \(\mathfrak g\). The family of Cayley transforms is parametrized by admissible linear forms on \(\mathfrak g\) (that is, elements of \(\mathfrak g ^*\) which together with an almost complex structure give \(\mathfrak g\) the structure of a normal \(j\)-algebra), which in turn may be described by positive \(r\)-tuples where \(r\) is the rank of the normal \(j\)-algebra \(\mathfrak g\). The actual construction of the Cayley transforms, which involves pseudoinverses of certain log convex functions on the cone \(\Omega\) associated with \(D\), is similar to that given by the author in previous work [Transform. Groups 6, 227-260 (2001; Zbl 1015.32019)]. The author explicitly computes these Cayley transforms in the case of Piatetskii-Shapiro's non-quasisymmetric 4-dimensional Siegel domain, and makes a comparison with \textit{L. Geatti}'s transforms [Rend. Math. 2, 475-497 (1982; Zbl 0522.32027)].
scientific article; zbMATH DE number 1970789
scientific article