Has V. Lafforgue proved the automorphic-to-Galois direction in the Global Langlands conjectures for general reductive groups over function fields? What is the current status, more generally?

Related What is the current status of the function fields Langlands conjectures?

  • 4
    $\begingroup$ Depends on what you mean by automorphic to Galois. He associates a Galois representation to an automorphic form, this is compatible with Satake at unramified primes. I don't think any compatibility statement at ramified primes has been proved. $\endgroup$
    – Will Sawin
    Aug 15 '17 at 18:40
  • $\begingroup$ To the OP: are you one and the same person as @user113452 who edited your question? I find it an improbable conincidence that you have consecutive user numbers and are active on the same question. $\endgroup$
    – Alex M.
    Aug 16 '17 at 9:33
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    $\begingroup$ No. He's my officemate, though. $\endgroup$
    – user113453
    Aug 16 '17 at 12:51

The abstract of V. Lafforgue's paper https://arxiv.org/abs/1404.6416 says

For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the direction "from automorphic to Galois".

Theorem 1.1 of this paper explains precisely what is meant by this in the split case, and section 2.1 does the same in the non-split case. In short, the space of cusp forms admits a decomposition indexed by global Langlands parameters, compatible with Satake at unramified primes.

In addition to the reverse direction (i.e. the Arthur multiplicity formula), Lafforgue also mentions some further work that might be considered part of the Langlands correspondence:

I think the compatibility with the local Langlands correspondence is still work in preparation of Genestier and Lafforgue.

I think the compatibility with the theta correspondence is still unproved.

  • $\begingroup$ Is it possible to give a flavour of what the Arthur multiplicity formula says? $\endgroup$
    – Dr. Evil
    Apr 3 '20 at 4:32
  • $\begingroup$ @Dr.Evil I'm not really an expert on this, but the Arthur multiplicity formula describes how many automorphic form correspond to a given Galois representation, and which automorphic representations they appear in. In $GL_n$, the answer is on automorphic form in one representation, as long as the Galois representations have the purity properties that mean we expect it to come from an automorphic form at all. For general groups $G$, the answer is more complicated, and involves calculating with representations of the centralizer in $G$ of the image of Galois. $\endgroup$
    – Will Sawin
    Apr 3 '20 at 13:00
  • $\begingroup$ Thanks for the reply. Can you remind us please what are the purity properties for GL_n? Also, what are the references for Arthur's multiplicity formula? Arthur's original papers are rather technical, so I'm hoping there is a review article out there, skipping the proofs but giving the basic ideas. Also, Arthur is usually dealing with number fields. I imagine the situation is simpler over function fields. Is that explained somewhere? $\endgroup$
    – Dr. Evil
    Apr 4 '20 at 3:12

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