On Poincare series of filtrations on equivariant functions of two variables

Abstract

Let a finite group $G$ act on the complex plane $({\mathbb{C}}^2,0)$. We consider multi-index filtrations on the spaces of germs of holomorphic functions of two variables equivariant with respect to 1-dimensional representations of the group $G$ defined by components of a modification of the complex plane ${\mathbb{C}}^2$ at the origin or by branches of a $G$-invariant plane curve singularity $(C,0)\subset ({\mathbb{C}}^2,0)$. We give formulae for the Poincare series of these filtrations. In particular, this gives a new method to obtain the Poincare series of analogous filtrations on the rings of germs of functions on quotient surface singularities.