Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local system (=lisse sheaf) on $X-S$, where $S$ is a finite set of points. Then there exists a Hecke eigensheaf on the moduli stack $\mathrm{Bun}_n$, of rank $n$ vector bundles on $X$ equipped with appropriate level structures on $S$, whose Hecke eigenvalue is $\mathcal{L}$.
For $n=1$, this amounts to the global class field theory, so I will restrict to the case $n>1$. If $S$ is empty then this statement was proved for $n=2$ by Drinfeld and for arbitrary $n$ by Frenkel--Gaitsgory--Vilonen. If $S$ is non-empty (i.e. $\mathcal{L}$ is ramified) then much less is known. For instance, I know of only one case when this statement has been established in presence of wild singularities. This is when $X=\mathbb{P}^1$ and $\mathcal{L}$ is the generalised Kloosterman sheaf (which has wild singularity at $\infty$ and tame singularity at $0$). The above statement in this case was proved by Heinloth--Ngo--Yun.
I appreciate it if somebody can point out a reference where the conjecture is precisely formulated and also if people can list all the known cases.
------------------------ Added in edit:
Here are bolder assertion:
**) The same statement is expected to hold if we consider all local systems (not just irreducible ones).
***) There is a bijection between local systems on open dense subsets of $X$ and (equivalence classes of) Hecke eigensheaves on $\mathrm{Bun}_n$ (equipped with appropriate level structure at the ramified points).
-------------------- Added in further edit: As David Ben-Zvi points out, ** and *** are not precise enough as they don't take into account Arthur's parameters. How about the following statement:
*+) There is a canonical bijection between irreducible rank $n$ local systems on open dense subsets of $X$ and (equivalence classes of) cuspidal Hecke eigensheaves on $\mathrm{Bun}_n$ (equipped with appropriate level structure at the ramified points).
At the decategorified level, this would give the classical (i.e. non-geometric) bijection between irreducible Galois representations and cuspidal automorphic representations, proved by Drinfeld and L. Lafforgue.
Of course, to make any of this precise one has to specify what one means by a Hecke eigensheaf in the ramified situation. One place this is (roughly) discussed is in the introduction of Heinloth's PhD thesis. Another place is in Section 4.3 of Yun's CDM paper, though the latter also includes an "automorphic data" in the definition (which we may or may not want).
------------More precise formulation:
As Will and David point out, one needs to specify what one really means by Hecke eigensheaf and by level structures before one can judge if *+ is reasonable. It is not clear to me how to handle this. But here is an attempt.
Let $K$ be a compact open subgroup of $\mathrm{GL}_n(\mathbb{A})$. Then, I believe, we do know what a Hecke eigensheaf on $\mathrm{GL}_n(k(X))\backslash \mathrm{GL}_n(\mathbb{A})/K$ is. Let us call this a $K$-Hecke eigensheaf. Let us call a $K$-Hecke eigensheaf $E$ and $K'$-Hecke eigensheaf $E'$ equivalent if there exist a subgroup $K''\subset K\cap K'$ such that the pullbacks of $E$ and $E'$ to $\mathrm{GL}_n(k(X))\backslash \mathrm{GL}_n(\mathbb{A})/{K''}$ are isomorphic.
Now I guess conjecture *+ is that there is a bijection between irreducible local systems and equivalence classes of Hecke eigensheaves.
Note that one could (and perhaps should) define the equivalence using Yun's notion of automorphic data. This means that in addition to $K$, we also a choose a character $\chi$ of $K$.
A further intriguing alternative (suggested by Will Sawin) is to use the conductor. That might remove the ambiguity of this $K$.
