The global shtuka–Langlands cohomology conjecture

Assume that $G$ is split over ${\mathbb F}_q$, fix an open compact level $K\subset G({\mathbb O}_F)$, and let $Q$ be the set of places where $K_v\neq G({\mathcal O}_v)$. For a finite set $S$, let $\Sht_K(G)_{\Delta(\overline\eta)}$ be the diagonal fiber of the moduli stack of $G$-shtukas, let $V$ be a finite projective representation of $(\widehat G)^S$ over $\Lambda$, and let ${\mathfrak A}_{K_v}$ be the conjectural coherent sheaf on $\Loc_{\widehat G,v}$ attached to $K_v$. The global shtuka–Langlands cohomology conjecture. There is a natural $(W_{F,Q}^S\times H_{K,\Lambda})$-equivariant isomorphism

$$R\Gamma\bigl(\operatorname{Loc}_{\widehat G,Q}\otimes\Lambda, \widetilde V\otimes \operatorname{res}^!(\boxtimes_{v\in Q}{\mathfrak A}_{K_v})\bigr)\cong C_c\bigl(\operatorname{Sht}_K(G)_{\Delta(\overline\eta)},\operatorname{Sat}_S(V)\bigr).$$

This conjecturally identifies Galois-side coherent cohomology with compactly supported cohomology of global shtuka spaces, compatibly with Weil-group and derived Hecke actions.