Hausel–Letellier–Rodriguez-Villegas conjecture for generic character stacks

Let $\mathcal{C}$ be a generic $k$-tuple of semisimple conjugacy classes, let $\alpha_{\mathcal{C}}$ be its associated dimension vector, and let $\mathbb{H}_{\alpha_{\mathcal{C}}}(z,w)$ be the rational function associated with this dimension vector. The mixed Poincaré series is denoted by $H_c(\mathcal{M}_{\mathcal{C}},q,t)$. Hausel–Letellier–Rodriguez-Villegas conjecture. For any generic $k$-tuple $\mathcal{C}$ of semisimple conjugacy classes,

$$\dfrac{H_c(\mathcal{M}_{\mathcal{C}},q,t)}{(qt^2)^{-(\alpha_{\mathcal{C}},\alpha_{\mathcal{C}})}}=\dfrac{(qt^2)\mathbb{H}_{\alpha_{\mathcal{C}}}\left(-t\sqrt{q},\dfrac{1}{\sqrt{q}}\right)}{qt^2-1}.$$

This is the mixed-Poincaré refinement of the known E-series formula for generic character stacks and serves as the starting point for the paper’s proposed extension to non-generic semisimple monodromies.