On Poincaré series of filtrations on equivariant functions of two variables

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 the exceptional divisor 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 Poincaré series of these filtrations. In particular, this gives a new method to obtain the Poincaré series of analogous filtrations on the rings of germs of functions on quotient surface singularities.