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Journal | Annales Henri Poincare |
---|---|

Date | Accepted/In press - 31 Aug 2017 |

Date | E-pub ahead of print - 13 Oct 2017 |

Date | Published (current) - Jan 2018 |

Issue number | 1 |

Volume | 19 |

Number of pages | 69 |

Pages (from-to) | 71-139 |

Early online date | 13/10/17 |

Original language | English |

Let g be a semisimple Lie algebra over C. Let ν∈Autg be a diagram automorphism whose order divides T∈Z≥1. We define cyclotomic g-opers over the Riemann sphere P1 as gauge equivalence classes of g-valued connections of a certain form, equivariant under actions of the cyclic group Z/TZ on g and P1. It reduces to the usual notion of g-opers when T=1.

We also extend the notion of Miura g-opers to the cyclotomic setting. To any cyclotomic Miura g-oper ∇ we associate a corresponding cyclotomic g-oper. Let ∇ have residue at the origin given by a ν-invariant rational dominant coweight λˇ0 and be monodromy-free on a cover of P1. We prove that the subset of all cyclotomic Miura g-opers associated with the same cyclotomic g-oper as ∇ is isomorphic to the ϑ-invariant subset of the full flag variety of the adjoint group G of g, where the automorphism ϑ depends on ν, T and λˇ0. The big cell of the latter is isomorphic to Nϑ, the ϑ-invariant subgroup of the unipotent subgroup N⊂G, which we identify with those cyclotomic Miura g-opers whose residue at the origin is the same as that of ∇. In particular, the cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is interpreted as taking ∇ to other cyclotomic Miura g-opers corresponding to elements of Nϑ associated with simple root generators.

We motivate the introduction of cyclotomic g-opers by formulating two conjectures which relate them to the cyclotomic Gaudin model of [arXiv:1409.6937].

We also extend the notion of Miura g-opers to the cyclotomic setting. To any cyclotomic Miura g-oper ∇ we associate a corresponding cyclotomic g-oper. Let ∇ have residue at the origin given by a ν-invariant rational dominant coweight λˇ0 and be monodromy-free on a cover of P1. We prove that the subset of all cyclotomic Miura g-opers associated with the same cyclotomic g-oper as ∇ is isomorphic to the ϑ-invariant subset of the full flag variety of the adjoint group G of g, where the automorphism ϑ depends on ν, T and λˇ0. The big cell of the latter is isomorphic to Nϑ, the ϑ-invariant subgroup of the unipotent subgroup N⊂G, which we identify with those cyclotomic Miura g-opers whose residue at the origin is the same as that of ∇. In particular, the cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is interpreted as taking ∇ to other cyclotomic Miura g-opers corresponding to elements of Nϑ associated with simple root generators.

We motivate the introduction of cyclotomic g-opers by formulating two conjectures which relate them to the cyclotomic Gaudin model of [arXiv:1409.6937].

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