分圆Gaudin模型、Miura算子和flag簇
Cyclotomic Gaudin models, Miura opers and flag varieties
摘要：
设$\mathfrak{g}$是$\mathbb{C}$上的半单李代数.设$ν\in \text{Aut}\, \mathfrak{g}$是一个图自同构，其阶整除$T \in \mathbb{Z}_{\geq 1}$。我们将Riemann球面上的分圆$\mathfrak{g}$ -opers定义为某种形式的$\mathfrak{g}$值联络的规范等价类，在循环群$\mathbb{Z}/ T\mathbb{Z}$对$\mathfrak{g}$和$\mathbb{P}^1$的作用下等变。当$T = 1$时，它简化为通常的$\mathfrak{g}$ -opers概念。我们还将Miura $\mathfrak{g}$ -opers的概念推广到分圆的情形。对于任何分圆的Miura $\mathfrak{g}$ -oper $\nabla$，我们关联一个相应的分圆的$\mathfrak{g}$ -oper。设$\nabla$在原点有剩余，由$ν$ -不变有理支配余权$\checkλ_0$给出，并且在$\mathbb{P}^1$的覆盖上是无单值的.证明了与$\nabla$相同的分圆$\mathfrak{g}$-oper相关的所有分圆Miura $\mathfrak{g}$ -oper的子集同构于$\mathfrak{g}$的伴随群$G$的全标志簇的$\vartheta$ -不变子集，其中自同构$\vartheta$依赖于$ν$，$T$和$\checkλ_0$ .后者的大胞室同构于$N^\vartheta$，即幂幺子群$N \subset G$的$\vartheta$不变子群，我们将其与那些在原点处的残基与$\nabla$相同的分圆三浦$\mathfrak{g}$ -opers相鉴别。特别地，最近在[arXiv：1505.07582]中引入的分圆生成过程被解释为将$\nabla$带到与简单根生成元相关联的$N^\vartheta$的元素相对应的其他分圆Miura $\mathfrak{g}$ -opers。我们通过将它们与[arXiv：1409.6937]的分圆Gaudin模型相关联的两个构造来激励分圆$\mathfrak{g}$ -opers的引入。
英文摘要：
Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. Let $ν\in \text{Aut}\, \mathfrak{g}$ 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 $ν$-invariant rational dominant coweight $\checkλ_0$ 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 $ν$, $T$ and $\checkλ_0$. 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].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
