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Let $X$ be a projective curve and $G$ be a semisimple Lie group. There is a theorem roughly stating that there exists an isomorphism between the moduli space of principal $G$-bundles on $X$ and the moduli space of $G$-affine opers on $X$. Moreover, this isomorphism is compatible with the action of Heisenberg subgroup of the loop group $LG$ on the former and the action of generalized Drinfeld--Sokolov hierarchy on the latter.

For $X$ a hyperelliptic curve and $G=SL_2$, this theorem is essentially equivalent to Krichever's classical construction of algebro-geometric solutions of KdV hierarchy (and the compatibility condition expresses the fact that KdV flow linearizes on the Jacobian variety of $X$).

My question is: is there anything known about analogue of this theorem for $q$-difference operators (as opposed to opers, which are basically differential operators)? If such an analogue exists, it would presumably entail a $q$-version of Drinfeld--Sokolov hierachy.

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    $\begingroup$ There is a theory of q-Drinfeld-Sokolov reduction and q-analogues of opers, starting from arxiv.org/abs/q-alg/9704011 and arxiv.org/abs/q-alg/9702016, see also several subsequent papers of Sevostyanov where quantum group analogues of Whittaker reduction and [finite] W-algebras are studied. They arise in describing the center of quantum affine algebras, much as opers due for affine algebras in Feigin-Frenkel. $\endgroup$ Jun 19, 2018 at 8:50
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    $\begingroup$ I don't know that anyone has worked out an analogue of the theorem you mention for them - but a q-analogue of the Krichever construction exists, going back to work of van Moerbeke and Mumford, see Mumford's inspirational article "An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equation. " $\endgroup$ Jun 19, 2018 at 8:53
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    $\begingroup$ The description of the theorem you quote is somewhat misleading I think. One considers not just G-bundles but ones with data of a formal spectral curve. The isomorphism relates generic such beasts on P^1 with affine opers on the formal disc, not on P^1. Also the Krichever case comes from taking again X=P^1 but the formal spectral curve coming as the germ of a hyperelliptic curve. $\endgroup$ Jun 19, 2018 at 8:56
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    $\begingroup$ I should mention the main extension of the paper you mention that I'm aware of is Safronov's very nice paper arxiv.org/abs/1302.3540 on an algebro-geoemtric setting for string equations / Virasoro constraints. $\endgroup$ Jun 19, 2018 at 9:01
  • $\begingroup$ @DavidBen-Zvi thank you very much for your extensive comments, I will be more careful next time I quote this theorem $\endgroup$
    – user74900
    Jun 19, 2018 at 11:11

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