Cyclotomic Gaudin models, Miura opers and flag varieties

TY - JOUR

T1 - Cyclotomic Gaudin models, Miura opers and flag varieties

AU - Vicedo, Benoit

AU - Lacroix, Sylvain

N1 - © Springer International Publishing AG 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.

PY - 2018/1

Y1 - 2018/1

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].

U2 - 10.1007/s00023-017-0616-8

DO - 10.1007/s00023-017-0616-8

M3 - Article

VL - 19

SP - 71

EP - 139

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0661

IS - 1

ER -