Poincar\'{e} series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

Abstract

A relation is proved between the Poincar\'{e} series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type $A_{2n}$, this amounts to the statement that the Poincar\'{e} series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence.