The geometric Langlands conjecture for semisimple groups

Let $C$ be the curve, let $G$ and ${}^{L}G$ be a pair of Langlands-dual semisimple groups, let ${}^{L}\operatorname{\boldsymbol{\mathcal{B}un}}$ be the moduli stack of principal ${}^{L}G$-bundles on $C$, and let $\operatorname{\boldsymbol{\mathcal{L}oc}}$ be the moduli stack of algebraic $G$-local systems on $C$. For a sheaf of algebras $\mathcal{A}$ on an algebraic stack $X$, write $D_{\operatorname{coh}}(X,\mathcal{A})$ for the derived category of complexes of $\mathcal{A}$-modules with coherent cohomology sheaves.

The geometric Langlands correspondence. There exists a canonical equivalence of categories

$$\mathfrak{c}:D_{\operatorname{coh}}(\operatorname{\boldsymbol{\mathcal{L}oc}},\mathcal{O})\stackrel{\cong}{\longrightarrow}D_{\operatorname{coh}}({}^{L}\operatorname{\boldsymbol{\mathcal{B}un}},\mathcal{D}),$$

which intertwines the action of the tensorization functors on $D_{\operatorname{coh}}(\operatorname{\boldsymbol{\mathcal{L}oc}},\mathcal{O})$ with the action of the Hecke functors on $D_{\operatorname{coh}}({}^{L}\operatorname{\boldsymbol{\mathcal{B}un}},\mathcal{D})$.

This is the geometric Langlands correspondence in the formulation attributed in the source to Deligne, Laumon, and Beilinson–Drinfeld. Its validity is presented as a conjectural statement; the source does not provide evidence of resolution.