Bedford, E.; Dadok, J. Proper holomorphic mappings and real reflection groups. (English) Zbl 0556.32012 J. Reine Angew. Math. 361, 162-173 (1985). We consider proper holomorphic mappings \(f:\Omega \to D,\) where D is an arbitrary complex manifold and \(\Omega\) is one of the following: (1) a complexified n-sphere \(\Omega =\Sigma (R)=\{z\in {\mathbb{C}}^ n:z^ 2_ 1+...+z^ 2_{n+1}=1,\| z\| <R\};\) (2) an incomplete Reinhardt domain. In both cases, the set of all possible proper mappings f is described. - In case (1), f is biholomorphically equivalent to the natural projection onto a quotient of \(\Sigma\) (R) by a finite subgroup of \(O(n+1)\). This subgroup is either fixed point free or a real reflection group. In case (2), it is shown that there exists an unbranched covering map \(f_ 1:\Omega \to \Omega_ 1\) to another incomplete Reinhardt domain \(\Omega_ 1\); and f may be factored as \(f=f_ 2f_ 1\), where \(f_ 2: \Omega_ 1\to D\) is biholomorphic to a quotient of \(\Omega_ 1\) by a finite group \(\Gamma\) of automorphisms. The possible groups \(\Gamma\) and the resulting mappings \(f_ 2\) are analyzed in some detail. Cited in 6 Documents MSC: 32H99 Holomorphic mappings and correspondences 32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) 51F15 Reflection groups, reflection geometries Keywords:proper holomorphic mappings; complexified n-sphere; incomplete Reinhardt domain; real reflection group PDFBibTeX XMLCite \textit{E. Bedford} and \textit{J. Dadok}, J. Reine Angew. Math. 361, 162--173 (1985; Zbl 0556.32012) Full Text: Crelle EuDML