Proper holomorphic maps on an irreducible bounded symmetric domain of classical type (Q909842)

Let B be an irreducible bounded symmetric domain of classical type and of dimension \(n>1\). Let \(\Omega\) be a domain in \({\mathbb{C}}^ n\), and let F: \(B\to \Omega\) be a proper holomorphic mapping of multiplicity \(m>1\). Suppose F has a \({\mathcal C}^ 1\) extension to the closure of B. The author proves that there exists a finite group G of linear transformations of \({\mathbb{C}}^ n\) generated by reflections, a polynomial mapping P: \({\mathbb{C}}^ n\to {\mathbb{C}}^ n\), two biholomorphic mappings \(\psi\) and \(\phi\) whose range and domain, respectively, is the unit ball in \({\mathbb{C}}^ n\) with respect to a G-invariant norm, such that \(F=\phi \circ P\circ \psi\). This result generalizes earlier work due to \textit{W. Rudin} [Indiana Univ. Math. J. 31, 701-720 (1982; Zbl 0506.32009)].