Polynomials associated to Lie algebras

We associate to a semisimple complex Lie algebra $\mathfrak{g}$ a sequence of polynomials $P_{\mathrm{\ell },\mathfrak{g}}(x)\in \mathbb{Q}[x]$ in $r$ variables, where $r$ is the rank of $\mathfrak{g}$ and $\mathrm{\ell }=0,1,2,\dots$. The polynomials $P_{\mathrm{\ell },\mathfrak{g}}(x)$ are uniquely associated to the isomorphism class of $\mathfrak{g}$, up to re-numbering the variables, and are defined as special values of a variant of Witten’s zeta function. Another set of polynomials associated to $\mathfrak{g}$ were defined in 2008 by Komori, Matsumoto and Tsumura using different special values of another variant of Witten’s zeta function.