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Let $N\unlhd G$ be a pair of finite subgroups of $\mathrm{SL}_2(\mathbb{C})$and $V$ a finite-dimensional fundamental $G$-module. We study Kostant'sgenerating functions for the decomposition of the $\mathrm{SL}_2(\mathbbC)$-module $S^k(V)$ restricted to $N\lhd G$ in connection with theMcKay-Slodowy correspondence. In particular, the classical Kostant formula wasgeneralized to a uniform version of the Poincar\'{e} series for the symmetricinvariants in which the multiplicities of any individual module in thesymmetric algebra are completely determined.

Kostant's generating functions and McKay-Slodowy correspondence