Invariant holomorphic mappings

Given a real-analytic hypersurface invariant under a finite unitary group, we construct an invariant holomorphic mapping to a hyperquadric, and prove the basic properties of this mapping. When the hypersurface is the unit sphere, the groups are cyclic, and the quotient is a Lens space, we prove that the coefficients of this mapping must be square roots of integers. For the Lens spaces L(p, p - 1) we evaluate these integers by some combinatorial reasoning. We indicate how these calculations bear on a conjecture about the multiplicity of proper mappings between balls in different dimensions.