Kostant's generating functions and McKay-Slodowy correspondence

TL;DR

This paper explores Kostant's generating functions for module decompositions related to finite subgroups of SL_2(C), extending classical formulas and connecting to the McKay-Slodowy correspondence to determine module multiplicities.

Contribution

It generalizes Kostant's classical formula to a uniform Poincaré series for symmetric invariants, linking module multiplicities with the McKay-Slodowy correspondence.

Findings

Generalized Kostant's formula to a uniform Poincaré series.

Connected module decompositions to the McKay-Slodowy correspondence.

Determined multiplicities of modules in symmetric algebras.

Abstract

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's generating functions for the decomposition of the $\mathrm{SL}_2(\mathbb C)$-module $S^k(V)$ restricted to $N\lhd G$ in connection with the McKay-Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincar\'{e} series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.