The geometric Breuil–Mézard conjecture for general linear groups

Let $G=\operatorname{GL}_n$. For each Serre weight $\sigma$ of $G(\mathbf{F}_p)$, let $\mathcal{Z}(\sigma)$ be a cycle in the $d$-dimensional cycle group of the reduced Emerton–Gee stack, where $d=\dim\check{\mathcal{B}}$ is the dimension of the flag variety of the dual group. For a dominant weight $\lambda$, an inertial parameter $\tau$, and the Jordan–Hölder multiplicity $n_\sigma(\lambda,\tau)=[\overline{W(\lambda)\otimes\sigma(\tau)}:\sigma]$, Emerton–Gee's geometric Breuil–Mézard conjecture. There is a collection of effective cycles $\mathcal{Z}(\sigma)\in Z_d((\mathcal{X}_{{}^{L}\!G}^{\operatorname{EG}})_{\operatorname{red}})$ such that

$$[\mathcal{X}^{\lambda+\rho,\tau}_{{}^{L}\!G,\mathbf{F}}]=\sum_\sigma n_\sigma(\lambda,\tau)\mathcal{Z}(\sigma)\in Z_d((\mathcal{X}_{{}^{L}\!G}^{\operatorname{EG}})_{\operatorname{red}}).$$

This refines the original Breuil–Mézard conjecture by asserting identities of cycles on the Emerton–Gee stack, rather than only numerical multiplicity formulas. The statement is presented for $\operatorname{GL}_n$ and is attributed to Emerton and Gee.