Poincaré series, exponents of affine Lie algebras, and McKay-Slodowy correspondence

Abstract

Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincaré series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)={\oplus }_{k\ge 0}{V}^{\otimes k}$ are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincaré series associated with the distinguished pairs of subgroups of ${\mathrm{SU}}_2$ give rise to the exponents of all untwisted and twisted affine Lie algebras except ${\mathrm{A}}_{2n}^{(1)}$.

22 pages