A McKay correspondence for the Poincaré series of some finite subgroups of SL_3(C)

Journal of Singularities
volume 18 (2018), 397-408

Received: 21 December 2017. Accepted: 14 June 2018.

DOI: 10.5427/jsing.2018.18t

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Abstract:

A finite subgroup of SL_2(C) defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups G of SL_3(C). They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that we obtain a similar relation between the group G and the corresponding surface singularity.

Mathematical Subject Classification:

32S25, 14E16, 13A50, 20G05

Author(s) information:

Wolfgang Ebeling
Institut für Algebraische Geometrie
Leibniz Universität Hannover
Postfach 6009, D-30060 Hannover, Germany
email: ebeling@math.uni-hannover.de