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arXiv 2007.087122020-07math.AGmath.RT

Hessenberg理想纤维的仿射铺砌

Affine Pavings of Hessenberg Ideal Fibers

Ke Xue

备注：: [v4] adds recent related works and citations of this paper. [v3] improves Lemma 2.5 to a new version that achieves the same result from a weaker assumption. Proposition 2.13 is remade to show that Lemma 2.5 still applies to $\mathcal{O}_v$ even though it is not necessarily a vector bundle over $\mathcal{O}_v^λ$. [v2] added an accidentally omitted case of $U$ for type $F_4$ in section 6

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AI总结：

本文定义了旗流形的闭子簇Hessenberg理想纤维并证明其由仿射铺砌，在$G_2$型中显式描述这些纤维，据此完全分类了Weyl群在正则半单Hessenberg簇上同调的Tymoczko点作用。

摘要：

我们定义了旗流形的某些闭子簇，即Hessenberg理想纤维，并证明它们由仿射铺砌。Hessenberg理想纤维是Springer纤维的自然推广。在$G_2$型中，我们显式描述了所有Hessenberg理想纤维，研究了它们的一些几何性质，并利用它们完全分类了Weyl群在正则半单Hessenberg簇上同调的Tymoczko点作用。

英文摘要：

We define certain closed subvarieties of the flag variety, Hessenberg ideal fibers, and prove that they are paved by affines. Hessenberg ideal fibers are a natural generalization of Springer fibers. In type $G_2$, we give explicit descriptions of all Hessenberg ideal fibers, study some of their geometric properties and use them to completely classify Tymoczko's dot actions of the Weyl group on the cohomology of regular semisimple Hessenberg varieties.

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