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Let $G$ be a complex, connected semi-simple Lie group, $G'$ its Langlands dual group, $\mathrm{Bun}_G$ the moduli stack of $G$-bundles on a smooth projective curve $\Sigma$ over complex numbers, $\mathrm{Loc}_{G'}$ the moduli stack of flat $G'$-bundles on $\Sigma$. Beilinson and Drinfeld have constructed an equivalence between the category of coherent sheaves on $\mathrm{Loc}_{G'}$ supported scheme-theoretically at the locus of opers and the category of $D$-modules on $\mathrm{Bun}_G$ admitting a certain global presentation (description copied from Frenkel-Teleman).

Is there some nice way to think about Beilinson-Drinfeld's result from the point of view of arithmetic Langlands?

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  • $\begingroup$ I find the local version of this story a little more transparent: we have a (categorical) representation of the loop group for every oper on the punctured disk. This is because the universal enveloping algebra of the affine Lie algebra has a large center equivalent to the algebra of functions on the space of opers, and so for every oper we get a character of this center. Then our categorical representation is just the category of affine Lie algebra modules where the center acts by this character. $\endgroup$
    – dhy
    Jun 1, 2019 at 3:19
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    $\begingroup$ The arithmetic analogue would be a simple construction that takes in some kind of object (a galois rep/motive with extra structure) and spits out a representation of a p-adic group. Looking at this description, it is evident what is missing in the arithmetic setting: an analogue of the category of modules over the affine Lie algebra. This is probably the most important tool available in the geometric setting that so far does not seem to have been transferred to arithmetic. $\endgroup$
    – dhy
    Jun 1, 2019 at 3:22

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