幂零轨道覆盖上的函数与双有理几何

中文摘要 AI 辅助

我们利用Springer分解的类似物来描述$G = SL_n$的任意幂零轨道的万有覆盖$\tilde{\co}$上的正则函数环的$G$-模结构。在扩展Springer分解的先前工作基础上，我们构造了一个簇$\tilde{\mcM}$，它在部分旗簇$G/P$的余切丛上有限，并且在$\tilde{\co}$的仿射化$\mcM$上真且双有理。我们使用双有理几何技术证明$\tilde{\mcM}$具有有理奇点，这提供了所需的同调消失性，从而将$\tilde{\co}$上的函数环描述为$G$的Levi子群的诱导表示。我们的结果还给出了$R(\tilde{\co})$作为分次$G$-模的结构描述。我们描述了$\mcM$的最小嵌入，研究了$\tilde{\co}$的分量群的特征到抛物子群和Levi子群的提升，并提出了一个更一般的消失猜想。

英文摘要

We use an analogue of the Springer resolution to describe the $G$-module structure on the ring of regular functions on the universal cover $\widetilde{\mathcal{O}}$ of any nilpotent orbit for $G = SL_n$. Building on previous work on the extended Springer resolution, we construct a variety $\widetilde{\mathcal{M}}$ that is finite over the cotangent bundle of a partial flag variety $G/P$, and proper and birational over the affinization $\mathcal{M}$ of $\widetilde{\mathcal{O}}$. We use techniques in birational geometry to show that $\widetilde{\mathcal{M}}$ has rational singularities, which provides the cohomology vanishing needed to describe the ring of functions on $\widetilde{\mathcal{O}}$ as an induced representation from a Levi subgroup of $G$. Our results also yield a description of the structure of $R(\widetilde{\mathcal{O}})$ as a graded $G$-module. We describe the minimal embedding of $\mathcal{M}$, study the lifting of characters of the component group of $\widetilde{\mathcal{O}}$ to parabolics and Levi subgroups, and make a more general vanishing conjecture.

发表机构

• University of Georgia（佐治亚大学）
• College of Charleston（查尔斯顿学院）
• University of Glasgow（格拉斯哥大学）

机构由 AI 辅助整理，请以论文原文为准。