In this paper, we carry out several computations involving graded (or ${{\mathbb {G}}_{\textrm {m}}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the ${{\mathbb {G}}_{\textrm {m}}}$-action on certain normalized (or ‘canonical’) simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse-coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the cohomology of tilting modules for the corresponding quantum group, as well as a proof that $\textsf {PCoh}^{{{\mathbb {G}}_{\textrm {m}}}}({\mathcal {N}})$ never admits a positive grading when the characteristic of the field is greater than 3.

中文翻译：

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梯度正交相干滑轮的计算

在本文中，我们进行了一些计算，这些计算涉及具有良好特征的还原群的幂等锥上的梯度（或$ {{\ mathbb {G}} _ {\ textrm {m}}} $-等变） 。在本文的第一部分中，我们计算了$ {{\ mathbb {G}} _ {\ textrm {m}}} $$操作在某些规范化（或“规范”）简单对象上的权重，确认了一个旧的奥斯特里克的预测。在本文的第二部分中，我们明确描述了除2或3以外的每个特性中，所有简单的正交相干滑轮$ G = PGL_3 $。应用包括对相应量子组的倾斜模的同调性的明确描述，以及$ \ textsf {PCoh} ^ {{{\ mathbb {G}} _ {\ textrm {m}}}}}（{\ mathcal {N}}）$的证明时，的字段大于3。