Kostant’s generating functions and Mckay–Slodowy correspondence

Let $N⊴G$ be a pair of finite subgroups of ${\mathrm{\text{SL}}}_2(ℂ)$ and $V$ a finite-dimensional fundamental $G$-module. We study Kostant’s generating functions for the decomposition of the ${\mathrm{\text{SL}}}_2(ℂ)$-module $S^k(V)$ restricted to $N◃G$ in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.