Cyclotomic Gaudin Models, Miura Opers and Flag Varieties

Abstract

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. Let $\nu \in \text{Aut}\, \mathfrak{g}$ be a diagram automorphism whose order divides $T \in \mathbb{Z}_{\geq 1}$. We define cyclotomic $\mathfrak{g}$-opers over the Riemann sphere $\mathbb{P}^1$ as gauge equivalence classes of $\mathfrak{g}$-valued connections of a certain form, equivariant under actions of the cyclic group $\mathbb{Z}/ T\mathbb{Z}$ on $\mathfrak{g}$ and $\mathbb{P}^1$. It reduces to the usual notion of $\mathfrak{g}$-opers when $T = 1$. We also extend the notion of Miura $\mathfrak{g}$-opers to the cyclotomic setting. To any cyclotomic Miura $\mathfrak{g}$-oper $\nabla$ we associate a corresponding cyclotomic $\mathfrak{g}$-oper. Let $\nabla$ have residue at the origin given by a $\nu$-invariant rational dominant coweight $\check{\lambda}_0$ and be monodromy-free on a cover of $\mathbb{P}^1$. We prove that the subset of all cyclotomic Miura $\mathfrak{g}$-opers associated with the same cyclotomic $\mathfrak{g}$-oper as $\nabla$ is isomorphic to the $\vartheta$-invariant subset of the full flag variety of the adjoint group $G$ of $\mathfrak{g}$, where the automorphism $\vartheta$ depends on $\nu$, $T$ and $\check{\lambda}_0$. The big cell of the latter is isomorphic to $N^\vartheta$, the $\vartheta$-invariant subgroup of the unipotent subgroup $N \subset G$, which we identify with those cyclotomic Miura $\mathfrak{g}$-opers whose residue at the origin is the same as that of $\nabla$. In particular, the cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is interpreted as taking $\nabla$ to other cyclotomic Miura $\mathfrak{g}$-opers corresponding to elements of $N^\vartheta$ associated with simple root generators. We motivate the introduction of cyclotomic $\mathfrak{g}$-opers by formulating two conjectures which relate them to the cyclotomic Gaudin model of [arXiv:1409.6937].