Unitary reflection groups and automorphisms of simple hypersurface singularities (Q2754595)

In the paper [Russ. Math. Surv. 54, No. 5, 873-893 (1999); translation from Usp. Math. Nauk 54, No. 5, 3-24 (1999; Zbl 0971.32010)] generalizing the Arnol'd approach to boundary singularities, the author studied smoothings of simple hypersurfaces invariant under a unitary reflection of finite order. It was observed that the monodromy on each of the character subspaces \(H_\chi\) is a finite group generated by unitary reflections. This way unitary reflection groups, \(G(m,1,k)\) and seven exceptional groups, made their first appearance in singularity theory addressing one of the long-standing questions of Arnol'd on realizations of the Shephard-Todd groups. Later, in [\textit{V. V. Goryunov} and \textit{C. E. Baines} [St. Petersburg Math. J. 11, No. 5, 761-774 (2000); translation from Algebra Anal. 11, No. 5, 74-91 (2000; Zbl 1002.58018)], some other Shephard-Todd groups were shown to be the monodromy groups of simple function singularities equivariant with respect to the action of finite order elements of \(SU(2)\). NEWLINENEWLINENEWLINEIn this paper, the author studies arbitrary finite order automorphisms of the zero levels of simple function-germs whose action can be extended to some of the smoothings. The monodromy in the space of the symmetric smoothings is still a Shephard-Todd group. The group \(G_{10}\) and two other realizations of the series \(G(m,1,k)\) are obtained. For the cases not considered earlier, the author constructs distinguished sets of generators of the \(H_\chi\). It is shown how symmetric Dynkin diagrams of the simple functions can be folded into diagrams of the equivariant singularities. The skew-Hermitian analogues of the unitary reflection groups under consideration are also described. NEWLINENEWLINENEWLINEThe paper is finished by giving a singularity theory interpretation of the rank 2 groups \(G_{12}\), \(G_{20}\) and \(G_{22}\), and showing how their Dynkin diagrams can be obtained by folding those of \(E_6\) and \(E_8\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00034].