Research output: Contribution to journal › Article › peer-review

**Cyclotomic Gaudin models, Miura opers and flag varieties.** / Vicedo, Benoit; Lacroix, Sylvain.

Research output: Contribution to journal › Article › peer-review

Vicedo, B & Lacroix, S 2017, 'Cyclotomic Gaudin models, Miura opers and flag varieties', *Preprint*. <http://eprints.whiterose.ac.uk/126067/>

Vicedo, B., & Lacroix, S. (2017). Cyclotomic Gaudin models, Miura opers and flag varieties. *Preprint*. http://eprints.whiterose.ac.uk/126067/

Vicedo B, Lacroix S. Cyclotomic Gaudin models, Miura opers and flag varieties. Preprint. 2017 Oct 13.

@article{09891e017714472a9d9059fa7220c25f,

title = "Cyclotomic Gaudin models, Miura opers and flag varieties",

abstract = "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].",

author = "Benoit Vicedo and Sylvain Lacroix",

year = "2017",

month = oct,

day = "13",

language = "English",

journal = "Preprint",

publisher = "Universit{\"a}t Potsdam",

}

TY - JOUR

T1 - Cyclotomic Gaudin models, Miura opers and flag varieties

AU - Vicedo, Benoit

AU - Lacroix, Sylvain

PY - 2017/10/13

Y1 - 2017/10/13

N2 - 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].

AB - 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].

M3 - Article

JO - Preprint

JF - Preprint

ER -