Invariant holomorphic mappings (Q1387670)

We investigate holomorphic mappings that are invariant under a finite subgroup of the unitary group on complex Euclidean space. Given such a group \(\Gamma\), and a real-analytic hypersurface \(M\) invariant under \(\Gamma\), we consider what hypersurfaces are possible targets for a non-constant \(\Gamma\)-invariant holomorphic mapping. We construct from a defining function \(r\) for \(M\) a \(\Gamma\)-invariant real-valued real-analytic function \(\Phi_{\Gamma,r}\), and use it to define an invariant holomorphic mapping taking \(M\) to a hyperquadric. We obtain general properties of \(\Phi_{\Gamma,r}\), and compute it for certain cyclic groups when \(M\) is the sphere \(S^3\). Its coefficients must then be integers; we determine them when \(S^3/ \Gamma\) is the lens space \(L(p,p-1)\), and find the target hyperquadrics. We also discuss a conjecture relating embedding dimensions of proper mappings between balls and the properties of invariant mappings.