Cyclotomic Gaudin models, Miura opers and flag varieties
In this paper, the authors define cyclotomic \(\mathfrak{g}\)-opers over the Riemann sphere, given a semisimple Lie algebra \(\mathfrak{g}\). Moreover, the authors extend the notion of Miura \(\mathfrak{g}\)-opers to the cyclotomic setting. They prove that the subset of all cyclotomic Miura \(\mathfrak{g}\)-opers associated with the same cyclotomic \(\mathfrak{g}\)-oper is isomorphic to the \(\theta\)-invariant subset of the full flag variety of the adjoint groups \(G\) of \(\mathfrak{g}\), where the authomorphism \(\theta\) is specified in terms of certain data. The introduction of cyclotomic \(\mathfrak{g}\)-opers is motivated by the two conjectures relating them to the cyclotomic Gaudin model of the second author and \textit{C. Young} [Commun. Math. Phys. 343, No. 3, 971--1024 (2016; Zbl 1383.17016)].
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