Given a semisimple complex linear algebraic group $G$ and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement ${𝒜}_I$, the regular nilpotent Hessenberg variety $\mathrm{Hess}(N,I)$, and the regular semisimple Hessenberg variety $\mathrm{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of ${𝒜}_I$ is isomorphic to $H^{*}\left(\mathrm{Hess}(N,I)\right)$ and $H^{*}{\left(\mathrm{Hess}(S,I)\right)}^W$, the invariants in $H^{*}\left(\mathrm{Hess}(S,I)\right)$ under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety $G/B$.

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黑森伯格的品种和超飞机安排

给定一个半简单复线性代数群 $G$在G的正根中有一个较低的理想I，出现了三个对象：理想排列${𝒜}_{\mathrm{一世}}$，常规的幂等Hessenberg变种 $\mathrm{赫斯}（ñ，\mathrm{一世}）$以及常规的半简单Hessenberg品种 $\mathrm{赫斯}（\mathrm{小号}，\mathrm{一世}）$。我们证明了从的对数导数模块导出的某个分级环${𝒜}_{\mathrm{一世}}$ 同构 $H^{*}（\mathrm{赫斯}（ñ，\mathrm{一世}））$ 和 $H^{*}{（\mathrm{赫斯}（\mathrm{小号}，\mathrm{一世}））}^{\mathrm{w\; ^}}$，其中的不变量 $H^{*}（\mathrm{赫斯}（\mathrm{小号}，\mathrm{一世}））$该Weyl群的作用下W¯¯的ģ。该同构表示为一般的Lie类型，并推广了Borel著名的定理，该定理表明W的协变代数与flag变种的同调环是同构的$G/乙$。