The local-global compatibility conjecture for mod $p$ representations of $\operatorname{GL}_n$

Let $\overline{r}:{\operatorname{Gal}}(\overline F/F)\rightarrow {\operatorname{GL}}_n(\mathbb{F})$ be a continuous homomorphism satisfying conditions (i)–(iii) in the paper, and let $v\mid p$ be a place of $F^+$ such that $F_v^+$ is unramified. Suppose $V^v\subseteq U^v$ are compact open subgroups with $V^v$ normal in $U^v$, let $\sigma^v$ be a finite-dimensional representation of $U^v/V^v$ over $\mathbb{F}$, and assume the indicated Hecke-isotypic space is nonzero. For $\widetilde v\mid v$, view this space as a representation of $G(F_{\widetilde v})\cong\operatorname{GL}_n(F_{\widetilde v})$. Local-global compatibility conjecture. There are an integer $d>0$ and an admissible smooth representation $\Pi_{\widetilde v}$ of $G(F_{\widetilde v})$ over $\mathbb{F}$ such that

$$\operatorname{Hom}_{U^v}(\sigma^v,S(V^v,\mathbb{F})[\mathfrak m^\Sigma])\cong\bigl(\Pi_{\widetilde v}\otimes(\omega^{n-1}\circ\det)\bigr)^{\oplus d},$$

where $\Pi_{\widetilde v}$ is compatible with one, equivalently any, good conjugate of $\overline r_{\widetilde v}$. This is the paper's main conjecture and predicts that the mod $p$ automorphic representation decomposes into local factors determined by the residual Galois representations; the stronger tensor-product formulation is stated separately.