Fargues–Scholze's categorical local Langlands conjecture

Let $F$ be a local field, let $G$ be a quasisplit reductive group over $F$, and let $B$ be a Borel subgroup with unipotent radical $N$. Choose a generic continuous character $\psi_v:N(F_v)\ra\La^\times$, and let $\LS_{\prescript{L}{}{G},F_v}$ be the moduli of continuous $\prescript{L}{}{G}$-valued representations of $W_{F_v}$. Let $c_{\psi_v}$ be the right adjoint to the colimit-preserving functor obtained from the spectral action on $i_{1,!}\!\cInd_{N(F_v)}^{G(F_v)}\psi_v$. Write $D(\Bun_{G,F_v},\La)^\om$ for the compact objects and $D_{\coh}^{\operatorname{qc}}(\LS_{\prescript{L}{}{G},F_v})_{\Nilp}$ for coherent sheaves with nilpotent singular support. Fargues–Scholze's categorical local Langlands conjecture. The functor $c_{\psi_v}$ restricts to an equivalence

$$D(\Bun_{G,F_v},\La)^\om\ra^\sim D_{\coh}^{\operatorname{qc}}(\LS_{\prescript{L}{}{G},F_v})_{\Nilp}.$$

Consequently, applying $\Ind$ yields an equivalence

$$\bL_{\psi_v}:D(\Bun_{G,F_v},\La)\ra^\sim\Ind D_{\coh}^{\operatorname{qc}}(\LS_{\prescript{L}{}{G},F_v})_{\Nilp}.$$

This is the categorical local Langlands conjecture of Fargues–Scholze. It is known when $G$ is a torus; the general quasisplit case remains open.