卢斯蒂格特殊片猜想
Lusztig's special pieces conjecture
中文总结 AI 辅助
研究简单代数群$G$的李代数$\mathfrak g$中特殊幂零轨道的特殊片，通过两种方法证明其是光滑$G$ - 簇被有限群作用的商，适用于经典$\mathfrak g$并给出新证，还涉及例外群及相关嵌入与定义。
中文摘要 AI 辅助
设$\mathcal O$为简单代数群$G$的李代数$\mathfrak g$中的一个特殊幂零轨道。我们给出两个证明，证明$\mathfrak g$中的每个特殊片${\mathcal P}(\mathcal O)$是某个光滑$G$ - 簇$X$被某个有限群$H$作用的商。首先从前三位作者和傅在早期工作中为横截切片建立的类似结果推导该结论。然后给出$X$的更明确构造，将其作为$\mathfrak g$与$G$的一些基本权表示的直和中一个$G$ - 轨道闭包的子簇。两种方法都适用于经典的$\mathfrak g$，在此我们给出该结果的新证明，此结果最初由克拉夫特和普罗塞西证明。例外群中的结果由卢斯蒂格猜想。第一个证明表明存在几个满足该猜想的$G$ - 簇$X$，与$H$在$\mathcal O$的基本群中的自然嵌入有关。在附录中，我们将此自然嵌入与卢斯蒂格对$H$的定义相关联，该定义源于$G$的外尔群中与$\mathcal O$相关的族以及斯普林格对应。
英文摘要
Let $\mathcal O$ be a special nilpotent orbit in the Lie algebra $\mathfrak g$ of a simple algebraic group $G$. We give two proofs of the result that every special piece ${\mathcal P}(\mathcal O)$ in $\mathfrak g$ is the quotient of a smooth $G$-variety $X$ by the action of a certain finite group $H$. We first deduce the result from a similar result for transverse slices, established in earlier work of the first three authors and Fu. Then we give a more explicit construction of $X$, as a subvariety of the closure of a $G$-orbit in the direct sum of $\mathfrak g$ and some fundamental weight representations of $G$. Both methods apply to classical $\mathfrak g$, where we give new proofs of this result, which was first proved by Kraft and Procesi. The result in the exceptional groups was conjectured by Lusztig. Our first proof shows that there can be several $G$-varieties $X$ that satisfy the conjecture, related to a natural embedding of $H$ in the fundamental group of $\mathcal O$. In an appendix, we relate this natural embedding to Lusztig's definition of $H$ that arises from the family in the Weyl group of $G$ attached to $\mathcal O$ and from the Springer correspondence.
