The refined geometric Breuil–Mézard conjecture for tame groups

Let $\sigma$ range over parahoric Serre weights, and let $\mathcal{X}^{\operatorname{crys},\lambda,\tau}_{{^{L}\!G},\mathbb{F}_p}$ have cycle class in the Chow group of $\mathcal{X}_{{^{L}\!G},\operatorname{red}}$. For a representation, write $[R:\sigma]$ for the multiplicity of $\sigma$ among its Jordan–Hölder factors. The refined geometric Breuil–Mézard conjecture. For each parahoric Serre weight $\sigma$, there exists an effective top-dimensional cycle $\mathcal{Z}_\sigma$ on $\mathcal{X}_{{^{L}\!G},\operatorname{red}}$ such that

$$[\mathcal{X}^{\operatorname{crys},\lambda,\tau}_{{^{L}\!G},\mathbb{F}_p}]=\sum_{\sigma}[\overline{\bar V(\operatorname{DL}^{-1}(\tau))\otimes V(\lambda-\eta_{\mathbb{Q}_p})}:\sigma]\mathcal{Z}_\sigma$$

for all regular $\lambda$ and tame inertial types $\tau$. This refinement predicts that the cycle classes of potentially crystalline stacks are governed by fixed cycles attached to Serre weights; it generalizes the cycle-theoretic Breuil–Mézard conjecture and remains open in this generality.