Poincaré series of binary polyhedral groups and McKay's correspondence (Q1097352)

Let G be a finite subgroup of \(SL_ 2({\mathbb{C}})\) and v be a complex irreducible character of G. Denote by \(m_{h,v}\) the multiplicity of v in the representation of G in the space of homogeneous polynomials of degree h in two variables. The Poincaré series \(P_ v(T)\) is given by the formula \[ P_ v(T)=\sum^{\infty}_{h=0}m_{h,v}T\quad h. \] The author gives explicit formulas for \(P_ v(T)\) for all v as rational functions. To do this he uses McKay's correspondence between finite subgroups of \(SL_ 2({\mathbb{C}})\) and Dynkin diagrams of affine type. See also the paper by \textit{B. Kostant} [in: Elie Cartan et les mathématiques d'aujourd'hui, Astérisque, No.Hors. Sér., 209-255 (1985; Zbl 0605.22010)].