The Breuil–Mézard cycle-map conjecture with fixed determinant

Let $K$ be a non-archimedean local field of residue characteristic $l$, let $\mathbb{F}$ be a finite field of characteristic $p\ne l$, and let $\overline\rho:G_K\rightarrow\operatorname{GL}_n(\mathbb{F})$ be continuous. Write $\mathtt{K}=\operatorname{GL}_n(\mathcal{O}_K)$, let $\operatorname{R}_{\mathbb{F}}(\mathtt{K})$ be its mod-$p$ representation ring, and let $\mathcal{Z}(\operatorname{R}_{\overline\rho}^{\square,\chi}/\lambda)$ denote the cycle group of the fixed-determinant deformation ring modulo $\lambda$. Breuil–Mézard cycle-map conjecture. There exists a unique mod-$\lambda$ cycle map

$$\overline{\operatorname{cyc}}:\operatorname{R}_{\mathbb{F}}(\mathtt{K})\longrightarrow\mathcal{Z}(\operatorname{R}_{\overline\rho}^{\square,\chi}/\lambda)$$

making the displayed reduction and component-map diagram commute. Uniqueness follows from surjectivity of the reduction map on the left. This is the cycle-map formulation of Breuil–Mézard for $l\ne p$ and is intended to connect representation-theoretic multiplicities with components of Galois deformation spaces; the paper establishes the corresponding function-field result.