Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

A relation is proved between the Poincaré 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 A2n, this amounts to the statement that the Poincaré 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. Introduction S. M. Gusein-Zade, F. Delgado, and A. Campillo [GDC] have shown that for an irreducible plane curve singularity the Poincaré series of the ring of functions on the curve coincides with the zeta function of its monodromy transformation. In this paper we show that there is also a relation between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. Let (X,x) be a normal surface singularity with good C-action. The coordinate algebra A is a graded algebra. We consider the Poincaré series pA(t) of A. Let {g; b; (α1, β1), . . . , (αr, βr)} be the orbit invariants of (X,x). We define ψA(t) := (1− t) r