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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 $K$.

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  • $\begingroup$ I think there are a few other examples, but not changing the basic point. (1) Either in HNY or in Yun's follow-up work, I believe one can handle the generalized Kloosterman sheaves of Katz (which have non-unipotent tame ramification at 0). (2) For GL_1 one can handle arbitrary wild ramification without much difficulty. $\endgroup$
    – Will Sawin
    Oct 17, 2019 at 14:19
  • $\begingroup$ Thanks Will. I edited the question, taking your comments into account. $\endgroup$
    – Dr. Evil
    Oct 18, 2019 at 13:25
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    $\begingroup$ @WillSawin In the categorical geometric setting (so char 0 & with D-modules) the strongest statement should be (modulo issues of non-temperedness) that $\operatorname{D-mod}(\operatorname{Bun}(X-S))$ and $\operatorname{QCoh}(\operatorname{LocSys}(X-S))$ correspond to each other under local categorical geometric Langlands. Of course, to extract concrete statements from this you need to pin down exactly what local geometric Langlands is. E.g., the knowledge that $\text{D-mod(Affine grassmannian)}$ corresponds under LGL to $\text{QCoh(pt/}\hat{G})$ recovers the unramified statement, the knowledge $\endgroup$
    – dhy
    Oct 19, 2019 at 2:29
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    $\begingroup$ that $\text{D-mod(Affine flag variety)}$ corresponds to $\operatorname{QCoh}(\hat{b}/\hat{B})$ recovers the tame (w unipotent monodromy) statements. There are even some correspondences which get into arbitrary ramification, e.g. that $g((t))-\operatorname{mod}_{crit}$ should correspond to $\text{QCoh(Opers)}$, and from that you can extract (rather painful) arbitrary ramification statements. Admittedly the original question was in the $l$-adic sheaf setting, where I believe that roughly the same story should hold but I am scared to write down any concrete statements... $\endgroup$
    – dhy
    Oct 19, 2019 at 2:39
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    $\begingroup$ But at the end of the day, one should be able to write down for every $\mathcal{L}$ an "explicit" category $C$ (described e.g. as a certain category of Whittaker sheaves) with a map from $C$ to the category of $\mathcal{L}$-eigensheaves. (And maybe one can do even more... but at this point already I would need to assume multiple wide open conjectures to even formulate my statements. And I am having difficulty expressing what I want to say in the space of the comment box.) $\endgroup$
    – dhy
    Oct 19, 2019 at 2:49

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