论文标题
约旦 - 椭圆形曲线上的$ g $捆的切瓦利分解
The Jordan--Chevalley decomposition for $G$-bundles on elliptic curves
论文作者
论文摘要
我们研究了$ 0 $ 0 $ nemuli $ g $ g $ $ g $ $ $ g $ e $ e $ e $ 1 $ 1 $ 1 $，其中$ g $是一个连接的还原群体。我们的主要结果描述了该堆栈的分区，由某个连接的还原子组$ h $ of $ g $（$ e $ -e $ -pseudo-levi子组）索引，其中每个层都以$ h $ bundles计算，以及相对Weyl组的作用。我们表明，此结果等于Jordan-Chevalley定理，用于在固定基础上配备框架的此类捆绑包。如果$ e $分别具有单个尖峰（分别为节点），则为lie代数$ \ mathfrak {g} $（分别，组$ g $）提供了Jordan-Chevalley定理的新证明。 我们还提供了Tannakian对这些模量堆栈的描述，并使用它表明，如果$ e $是普通的椭圆曲线，则在$ e $上的框架单体束的集合与$ g $的单体锥相同。最后，我们使用Borel-de siebenthal算法对$ e $ -PSEUDO-LEVI子组进行了分类，并计算一些明确的示例。
We study the moduli stack of degree $0$ semistable $G$-bundles on an irreducible curve $E$ of arithmetic genus $1$, where $G$ is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups $H$ of $G$ (the $E$-pseudo-Levi subgroups), where each stratum is computed in terms of $H$-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan--Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where $E$ has a single cusp (respectively, node), this gives a new proof of the Jordan--Chevalley theorem for the Lie algebra $\mathfrak{g}$ (respectively, group $G$). We also provide a Tannakian description of these moduli stacks and use it to show that if $E$ is an ordinary elliptic curve, the collection of framed unipotent bundles on $E$ is equivariantly isomorphic to the unipotent cone in $G$. Finally, we classify the $E$-pseudo-Levi subgroups using the Borel--de Siebenthal algorithm and compute some explicit examples.
