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I am trying to understand various ways in which one can modify the Langlands correspondence. Hopefully I will be able to learn something from you. First, one can categorify/decategorify.

It is my understanding the the geometric Langlands over complex numbers is by default categorical (yet there has been attempt to decategorify it by Frenkel and Langlands, see).

The arithmetic Langlands over finite extensions of $\mathbb{Q}_p$ is by default not categorical. I am not sure whether one can categorify it or not. Is there any literature on this?

For local fields of positive characteristic, I think the geometric Langlands and the arithmetic Langlands can co-exist (thus we have both a categorical statement and a non-categorical statement). Are there any implications from one to the other? A reference comparing the two in detail?

Second, one can also quantize. Gaitsgory is slowly but steadily understanding the "one-parameter" quantization of the geometric Langlands over complex numbers, it appears. There was a remark in Beilinson-Drinfeld about a certain "doubly quantized" local representation-theoretic result of Feigin-Frenkel, which they did not know how to globalize properly. This preprint appears to make first steps in that direction.

Can one also quantize or doubly quantize the usual arithmetic Langlands over finite extensions of $\mathbb{Q}_p$ or the function fields of positive characteristic? Can one quantize or doubly quantize the geometric Langlands in positive characteristic? Can one quantize or doubly quantize the de-categorified geometric Langlands over complex numbers? By the way, in the setting of Gaitsgory, the Langlands correspondence suddenly becomes more symmetrical than the non-quantized Langlands (i.e. you are computing "the same" categories, but with different parameters). Does something analogous happen here?

Possibly reasonable conjectures are hard to formulate for general reductive groups. What could the statements look like in the class field theory case, for example?

In all of the above variations, one can also ask how should the Arthur parameters (or the singular support condition, depending on what is more fundamental to you) look like. One can also ask how to incorporate ramification and how much ramification is to be allowed.

There is also one last way in which one can "modify" the Langlands correspondence but I am not sure it can even be properly called modification. In the local Langlands correspondence over $p$-adic numbers, you can either take the Galois representations with coefficients in $l$-adic numbers ($l \neq p$) or you can, in a pretty non-trivial fashion, take coefficients in $p$-adic numbers. What you get is the $p$-adic local Langlands correspondence. Can one $p$-adify any of the proposed correspondences above (where it makes sense)?

I realize that my questions are naive, I only wonder if it makes sense to ask them at all.

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You have asked a lot of questions all at once, but I will try to answer some of them.

1) A very minor point: I don't think it's quite right to say that Frenkel and Langlands attempt to decategorify it. I would say that Langlands is attempting to define a new, non-categorical correspondence and Frenkel is attempting to derive interesting non-categorical consequences from a piece of the categorical picture.

2) A more categorical version of the Langlands correspondence over $\mathbb Q_p$ was conjectured by Fargues and progress towards a proof was made by Scholze. Note that this is an analogue of the Betti geometric Langlands correspondence, so the category on the spectral side is simpler.

3) For global fields of positive characteristic, the geometric Langlands correspondence implies the classical Langlands correspondence in the direction Galois $\to$ automorphic. Indeed this is how Drinfeld proved classical Langlands for $GL_2$. You probably want to put "unramified" on both sides as I think the global ramified version is still not completely formulated. The proof of this is almost obvious from the definition, I think.

I think the same thing should work locally - given a local Galois representation, you should get a category with an action of the loop group and the action of Frobenius. Given such an object, you can define the Grothendieck group of the category of objects with an isomorphism to their Frobenius pullback. This has a bilinear form where you take the trace of Frobenius on $\operatorname{RHom}$ of two objects, and if you mod out by the kernel of this bilinear form you get a group, which you can view as a vector space over the coefficient field (where $\alpha$ acts by multiplying the Frobenius isomorphism by $\alpha$) with an action of the $\mathbb F_q$-points of the loop group, so this gives a representation of your algebraic group over your local field.

It is easy to see that this can be used to construct some interesting representations of local fields (induced representations, basically), but I don't know if this is worked out anywhere in the literature.

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  • $\begingroup$ Will, you say that Fargues has formulated a more categorical statement. Is it meaningful to ask for a more categorical statement of the $p$-adic local Langlands? Is it going to be the same statement of Fargues, or is it going to be something else entirely? $\endgroup$ – user141474 Jun 5 at 7:20
  • $\begingroup$ @DevaPath I don't know. If it's not mentioned in Fargues' paper then I would guess it's probably not known. $\endgroup$ – Will Sawin Jun 5 at 14:52

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