AI 中文总结
本文在正特征下将非临界水平 Kac-Moody 顶点代数与包络代数的 Harish-Chandra 中心等同于 Langlands 对偶群联络模空间上的函数代数，揭示了其非平凡且非特征零约化的新现象。
AI 中文摘要
设 $G$ 是定义在特征大于其单因子 Coxeter 数的域上的分裂约化群。我们将非临界水平下相关的 Kac-Moody 顶点代数和包络代数的 Harish-Chandra 中心与 Langlands 对偶群 $\check{G}$ 的联络模空间上的函数代数等同起来。具体而言，给定 $G$ 的一个非临界水平 $\kappa$，考虑定义在形式圆盘 $\mathscr{D}$ 上的相关 Kac-Moody 顶点代数 $V_\kappa(\mathfrak{g})$，及其弧群不变量的 Harish-Chandra 中心 $$V_\kappa(\mathfrak{g})^{\mathscr{J}G} \subset V_\kappa(\mathfrak{g}).$$ 记 $\check{\kappa}$ 为 $\check{G}$ 的对偶水平，$\check{\kappa}^p - \check{\kappa}$ 为其在 Artin-Schreier 映射下的像，$\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}$ 为 Frobenius 扭转圆盘 $\mathscr{D}^{(1)}$ 上 $(\check{\kappa}^p - \check{\kappa})$-opers 的模空间。我们建立了典范同构 $$\operatorname{Spec} V_\kappa(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}.$$ 对于水平 $\kappa$ 的滤过完备包络代数，其 Harish-Chandra 中心也有类似的等同，此时需将 $\mathscr{D}$ 替换为穿孔圆盘 $\mathscr{D}^\times$。这些 Harish-Chandra 中心的存在是正特征中非临界水平环路群的一个全新现象：这些中心是非平凡的，不同于特征零中非临界水平环路群的情形，并且特别地，它们不是特征零 Harish-Chandra 中心模 $p$ 的约化，这与有限维约化群或临界水平环路群的情形不同。
英文摘要
Let $G$ be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish--Chandra centers of the associated Kac--Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group $\check{G}$. Namely, given a noncritical level $κ$ for $G$, consider the associated Kac--Moody vertex algebra $V_κ(\mathfrak{g})$ defined on a formal disc $\mathscr{D}$, and its Harish--Chandra center of arc group invariants $$V_κ(\mathfrak{g})^{\mathscr{J}G} \subset V_κ(\mathfrak{g}).$$ Write $\checkκ$ for the dual level for $\check{G}$, $\checkκ^p - \checkκ$ for its image under the Artin--Schreier map, and $\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}$ for the moduli space of $(\checkκ^p - \checkκ)$-opers on the Frobenius twisted disc $\mathscr{D}^{(1)}$. We establish a canonical isomorphism $$\operatorname{Spec} V_κ(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}.$$ There is a similar identification for the Harish--Chandra center of the filtered complete enveloping algebra at level $κ$, where instead of $\mathscr{D}$ we need to consider the punctured disc $\mathscr{D}^\times$. The presence of these Harish--Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod $p$ of the characteristic zero Harish--Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.
Comments116 pages
