The Breuil–Mézard conjecture for potentially semistable deformation rings

Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}_K$ and residue field $k$, let $G_K$ be its absolute Galois group, and let $\overline{\rho}:G_K\rightarrow\operatorname{GL}_n(\mathbb{F})$ be continuous. For a regular Hodge–Tate weight $\lambda$ and an inertial Weil–Deligne type $\tau$, let $R_{\overline{\rho}}^{\lambda,\tau}$ be the reduced quotient of the framed deformation ring parametrizing potentially semistable lifts of type $(\lambda,\tau)$, and let $Z(R_{\overline{\rho}}^{\lambda,\tau}/\varpi)$ be its special-fiber cycle. For a virtual representation $V$ of $\operatorname{GL}_n(\mathcal{O}_K)$ over $E$, write $\overline V$ for its semisimple reduction modulo $\varpi$. Breuil–Mézard conjecture. There exist cycles $\mathcal{Z}_\sigma(\overline{\rho})$ in $\operatorname{Spec} R_{\overline{\rho}}^\square/\varpi$, indexed by irreducible $\operatorname{GL}_n(\mathcal{O}_K)$-representations $\sigma$ over $\mathbb{F}$, such that for every inertial type $\tau$ and regular $\lambda$,

$$Z(R_{\overline{\rho}}^{\lambda,\tau}/\varpi)=\sum_\sigma[\overline{r(\tau)\otimes_E V(\lambda-\eta)}:\sigma]\mathcal{Z}_\sigma(\overline{\rho}).$$

Here $r(\tau)$ is the virtual representation defined using inertial local Langlands, $V(\lambda-\eta)$ is the corresponding algebraic representation restricted to $\operatorname{GL}_n(\mathcal{O}_K)$, and the multiplicities may be negative. This conjecture expresses the non-reduced geometry of special fibers of potentially semistable deformation rings in terms of mod-$p$ representation theory; the paper studies related geometric and stack-theoretic formulations.