At its core, the Langlands program seeks to give a description of the vector space of automorphic forms. This is a space of functions on a locally symmetric space $X/\Gamma$, where $\Gamma=G(\mathbb Z)\subset G(\mathbb R)$ is an arithmetic group, equipped with its Hecke symmetries. One seeks a “dual” description, in terms of $L$-parameters, which are roughly representations of the absolute Galois group with values in the Langlands dual group.

Similar conjectures exist in the function-field case; in this setting, the quotient $X/\Gamma$ can be interpreted as the $\mathbb F_p$-points of the stack $\mathrm{Bun}_G$ of $G$-bundles on a curve $C$ over $\mathbb F_p$. Gaitsgory–Raskin, partly in large collaborations, proved a fine version of Langlands’ conjecture in this setting, in the everywhere unramified case. Notably, they define a vector space on the Galois side, as global sections of a sheaf on the moduli stack of $L$-parameters, and relate this to the space of automorphic forms, compatibly with the Hecke symmetries.

Their proof is a culmination of many decades of work in the geometric Langlands program, and consists of two key steps: 1) The space of automorphic forms is the trace of Frobenius on a category of $\ell$-adic sheaves on $\mathrm{Bun}_G$. 2) This category of $\ell$-adic sheaves on $\mathrm{Bun}_G$ is equivalent to a category of coherent sheaves on the stack of $L$-parameters. The starting point for 2) is the observation that this conjecture makes sense for curves over any field, and any sheaf theory, and is known as the geometric Langlands equivalence. The proof starts over the complex numbers, and uses $D$-modules.

[After Gaitsgory, Raskin, ...]