Abstract:
Finite groups generated by Euclidean reflections have been commonplace in various problems of singularity theory since their relationship with the classification of critical points of functions was discovered by Arnol'd [1], [2]. We show that a number of finite groups generated by unitary reflections are also naturally related to singularities of functions, namely, those invariant under a unitary reflection of finite order. To this end, we consider germs of functions on a manifold with boundary and lift them to a cyclic covering of the manifold, ramified over the boundary. This construction provides a new notion of roots for the groups under consideration and provides skew-Hermitian analogues of these groups.