For partitions $\lambda, \mu \vdash n$, the Kostka-Foulkes polynomial $K_{\lambda,\mu}(q)$, a $q$-analog of the Kostka coefficient $K_{\lambda,\mu}$, has a combinatorial description, due to Lascoux and Schützenberger, as $$ K_{\lambda,\mu}(q) = \sum_{T} q^{\mathrm{charge}(T)},$$ where the sum is over semistandard Young tableaux of shape $\lambda$ and content $\mu$. In the case of standard tableaux, charge is basically the same as major index (maybe we need to consider cocharge instead); at any rate, what is true is that if $\mu=(1,1,\ldots,1)$, then we have the $q$-hook length formula (due to Stanley) which gives a product formula for $K_{\lambda,\mu}(q)$: $$ K_{\lambda,\mu}(q) = \sum_{\textrm{$T$ a SYT of shape $\lambda$}} q^{\mathrm{maj}(T)}= \frac{[n]!_q}{\prod_{u\in\lambda} [h(u)]_q}.$$

The Kostka-Foulkes polynomials are the Type A version of Lusztig's $q$-analog of weight multiplicity (see papers of Lusztig and Joseph-Letzter-Zelikson cited below for the definition of these). As I understand it, giving a combinatorial description of these in other types as a statistical generating function over tableaux analogous to the charge formula is a pretty difficult problem. This problem is not fully resolved but the theory of crystals has led to significant progress (see the recent papers of Lecouvey-Lenart https://arxiv.org/abs/1707.03314 and Jang-Kwon https://arxiv.org/abs/1908.11041).

My question however is not about statistical generating functions, however. It is rather about product formulas.

**Question**: Are there any examples of Lusztig's $q$-analog of weight multiplicity, beyond the SYT $q$-hook length formula explained above, for which we have a product formula?

(Note that when we pass to the more general root system formulation, content $\mu=(1,1,\ldots,1)$ actually corresponds to weight $\mu=0$ since we mod out by $(1,1,\ldots,1)$ to form the weight lattice in Type A. But also note that not every Type A $q$-analog of weight multiplicity of the form $K_{\lambda,0}(q)$ has a hook-length formula because we also need that $\lambda$ is a partition with the right number of boxes.)

*Lusztig, George*, Singularities, character formulas, and a (q)-analog of weight multiplicities, Astérisque 101-102, 208-229 (1983). ZBL0561.22013.

*Joseph, Anthony; Letzter, Gail; Zelikson, Shmuel*, **On the Brylinski-Kostant filtration**, J. Am. Math. Soc. 13, No. 4, 945-970 (2000). ZBL0991.17006.

EDIT: I just realized that the question still makes sense even removing the "q" from it. That is...

**q=1 version of question**: Are there other product formulas for weight multiplicities of simple Lie algebras, beyond the Type A cases corresponding to SYTs?