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The Poincar\'{e} Series of a Quasihomogeneous Surface Singularity

The Poincaré series pA(t) of the coordinate algebra A of a normal surface singularity (X,x) with good C∗-action is written in a certain way as a quotient of two polynomials φA(t)/ψA(t). For a Kleinian singularity, we derive from the McKay correspondence that φA(t) and ψA(t) are the characteristic polynomials of the Coxeter element and the affine Coxeter element respectively. We show that if (X,x) is a hypersurface singularity or a certain ICIS then φA(t) is in a certain sense dual to the characteristic polynomial of the monodromy operator of the singularity. There are relations to the mirror symmetry of K3 surfaces and to automorphisms of the Leech lattice. Introduction K. Saito [Sa3, Sa4] has introduced a duality between polynomials which are products of cyclotomic polynomials. He has shown that V. I. Arnold’s strange duality between the 14 exceptional unimodal hypersurface singularities is related to such a duality between the characteristic polynomials of the monodromy operators of the singularities. Moreover, he has observed that the dual polynomials pair together to the characteristic polynomial of an automorphism of the Leech lattice. It is now well-known that Arnold’s strange duality is related to the mirror symmetry of K3 surfaces (see e.g. [D6]). The author [E4, E5] has shown that these features still hold in a certain way for the extension of Arnold’s strange duality discovered by C. T. C. Wall and the author [EW]. In this paper we consider a normal surface singularity (X,x) with good C-action. The coordinate algebra A is a graded algebra. We consider the Poincaré series pA(t) of A. If (X,x) is a Kleinian singularity, then we show that the Poincaré series pA(t) is a quotient pA(t) = φA(t)/ψA(t) where φA(t) and ψA(t) are the characteristic polynomials of the Coxeter element and the affine Coxeter element respectively of the corresponding root system. This is derived from the McKay correspondence using ideas of the paper [Kos]. There are various formulas for Poincaré series of binary polyhedral groups in [Kn, Kos, Sp], but this relation seems to be new. In the general case we also write pA(t) = φA(t)/ψA(t) as a quotient of two polynomials which may have common factors.