Finite irreducible groups, generated by reflections, are monodromy groups of suitable singularities
There is a well known correspondence between simple singularities (of a function of several complex variables) and finite reflection groups (of type A, D and E). A fundamental property of this is the existence of an isomorphism from the orbit space of the complexified action of the group to the miniversal deformation space of the singularity which (1) takes the variety of non-regular orbits to the discriminant, (2) induces an isomorphism of the reflection group with the monodromy group of the singularity, and (3) is induced by the period map. The object of this paper is to find analogues of these results for all finite reflection groups. For types B, C and F such analogues were already given by \textit{V. I. Arnol'd} [Usp. Mat. Nauk 33, No.5(203), 91-105 (1978; Zbl 0408.58009)]: the structure here is of a singularity of a function invariant under a group of order 2 acting by a reflection. For the rest, we have \(G_ 2(x^ 3+y^ 3,{\mathbb{Z}}_ 3);\) \(I_ 2(n)(x^ n+x^ 2y^ 2+y^ n,{\mathbb{Z}}_ n);\) \(H_ 3(x^ 5+y^ 5,{\mathbb{Z}}_ 5)\) where in each case \({\mathbb{Z}}_ n\) acts by \(\zeta (x,y)=(\zeta x,\zeta^{-1}y)\) \((\zeta^ n=1)\). The case of \(H_ 4\) is not treated here. The proofs of the assertions involve extensions of the techniques used in the cases previously known. A new feature is that as well as restricting the unfolding to one by invariant functions, the homology of the Milnor fibre is decomposed into eigenspaces for the group \({\mathbb{Z}}_ n\). Then for \(X=B_{\mu}\), \(C_{\mu}\), \(F_ 4\), \(I_ 2(n)\) and \(H_ 3\) the images of the vanishing cycles in the eigenspace belonging to \(\zeta (=\exp (2\pi i)/n)\) form a root lattice of type X, and the image of the monodromy group is the corresponding reflection group. (For the \(+1\) eigenspace we have a simpler reflection group in the cases B, C, F but an infinite one for \(I_ 2(n)\), \(H_ 3\). For \(G_ 2\) these cases are reversed.)
scientific article; zbMATH DE number 3914701
scientific article