Simple singularity of type E7 and the complex reflection group ST34

Abstract

This paper studies a family of surfaces of $C^3$ which is a deformation of a simple singularity of type $E_7$. This family has six parameters which are regarded as basic invariants of the complex reflection group No. 34 in the list of the paper of Shephard and Todd (1954). We compute 1-parameter subfamilies of the family in question corresponding to corank one reflection subgroups of No. 34 group. In particular, we determine the types of simple singularities on the surfaces appearing in this manner.