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Both over global function fields and $p$-adic fields, we have a series of conjectures under the name of “geometric Langlands conjectures”.

Over global function fields of char $p$, they are due to Drinfeld, L. Lafforgue and V. Lafforgue. Over $p$-adic fields, they are due to Fargues.

What about over the complex numbers, and over the real numbers?

Over $p$-adic fields $K$ with ring of integers $\mathcal{O}$, the formal affine line $\mathcal{O}[\![t]\!]$ plays a crucial role via Lubin–Tate theory.

In some geometric formulations of local Langlands over the complex numbers, it seems the punctured formal affine line also plays a crucial role.

My question is:

(1) is there a geometric Langlands correspondence, at least in conjectural form, for reductive groups over archimedean local fields?

(2) Does the formal affine line over an archimedean local field play a role in it, and if so, which role? Perhaps, somehow (how?), in analogy with local class field theory?

I'd appreciate some references.

Refs.

Mirkovic–Vilonen - Geometric Langlands duality and representations of algebraic groups over commutative rings

Frenkel - Lectures on the Langlands program and conformal field theory

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    $\begingroup$ Over C, see work of Edward Frenkel and Dennis Gaitsgory. E.g., Frenkel's book "Langlands Correspondence for Loop Groups". Over R, I recall seeing something by Nadler or Ben-Zvi. $\endgroup$
    – Marty
    Commented Jun 26, 2018 at 3:40
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    $\begingroup$ I think it's misleading to say that geometric Langlands ala Gaitsgory is a geometrization/categorification of complex archimedian Langlands. Rather, it's an analogue of nonarchimedian Langlands over the complex numbers. On the other hand, as I understand it, Ben-Zvi & Nadler's "Loop Spaces and Langlands Parameters" (which I assume is the paper Marty was thinking of) tries to explain archimedian Langlands as a S^1-equivariant localization of geometric Langlands, both in the real and complex cases. I'm secretly hoping David Ben-Zvi sees this and writes an answer... $\endgroup$
    – dhy
    Commented Jun 26, 2018 at 10:51
  • $\begingroup$ I agree with @dhy about the relationship between Frenkel-Gaitsgory and archimedean Langlands. I think I misunderstood the question a bit. Anyways, I think I should look at the paper of Ben-Zvi and Nadler! $\endgroup$
    – Marty
    Commented Jun 26, 2018 at 21:33

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