Towards a Combinatorial Model for $q$-weight Multiplicities of Simple Lie Algebras (Extended Abstract)

Koskta-Foulkes polynomials are Lusztig's q-analogues of weight multiplicities for irreducible representations of semisimple Lie algebras. It has long been known that these polynomials can be written with all non-negative coefficients. A statistic on semistandard Young tableaux with partition content, called charge, was used to give a combinatorial formula exhibiting this fact in type $A$. Defining a charge statistic beyond type $A$ has been a long-standing problem. Here, we take a completely new approach based on the definition of Kostka-Foulkes polynomials as an alternating sum over Kostant partitions, which can be thought of as formal sums of positive roots. This positive expansion in terms of Kostant partitions gives way to a statistic which is simply read by counting the number of parts in the Kostant partitions. The hope is that the simplicity of this new crystal-like model will naturally extend to other classical types.