This note is motivated by the problem of understanding Springer’s remarkable representation of the Weyl group W of a semisimple complex linear algebraic group G on the cohomology algebra of an arbitrary Springer variety in the ag variety of G from the viewpoint of torus actions and localization. Continuing the work [CK] which gave a sufficient condition for a group \( \mathcal{W} \) acting on the fixed point set of an algebraic torus action (S, X) on a complex projective variety X to lift to a representation of \( \mathcal{W} \) on the cohomology algebra H^*(X) (over ℂ), we describe when the representation on H^*(X) is equivalent to the representation of \( \mathcal{W} \) on the cohomology H^*(X^S) of the fixed point set. As a consequence of this theorem, we give a simple proof in type A of the Alvis–Lusztig–Treumann Theorem, which describes Springer’s representation of W for Springer varieties corresponding to nilpotents in a Levi subalgebra of Lie(G). Finally, we consider special torus actions (S, X), establishing sharp lower and upper bounds for the number of edges at each vertex of their moment graphs. We further show that a finite group \( \mathcal{W} \) acting on the moment graph 𝔐(X) of a special torus action acts on the cohomology algebra H^*(X) in two ways: namely the left and right or dot and star actions of Knutson [Knu] and Tymoczko [Tym]. It follows from our main theorem that the right action is induced from the natural representation of \( \mathcal{W} \) on H⁰(X^S).

中文翻译：

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色相作用下的环面作用，局部化和诱导表达

这说明是理解Springer的显着的Weyl群的代表性问题而导致的W¯¯半单复数线性代数群的摹对在股份公司各种任意斯普林格品种的上同调代数摹从花托行动和本地化的观点。继续该工作[CK]这给了一组的一个充分条件\（\ mathcal【W} \）作用在所述固定点集的代数环面的动作（的小号，X）上的复射影各种X到电梯的表示的\（\ mathcal【W} \）上的同调代数ħ ^*（X）（在over上），我们描述H ^*（X）上的表示等效于不动点集的同构H ^*（X ^S）上\（\ mathcal {W} \）的表示。作为该定理的结果，我们给出了Alvis–Lusztig–Treumann定理的类型A的简单证明，该定理描述了Lie（G）的Levi子代数中对应于幂幂的Springer品种的W的Springer表示。最后，我们考虑特殊的圆环动作（S，X），为矩量图每个顶点处的边数确定了清晰的上下限。我们进一步证明，作用于特殊圆环作用的矩图𝔐（X）上的有限群\（\ mathcal {W} \）以两种方式作用于同调代数H ^*（X）：即左和右或Knutson [Knu]和Tymoczko [Tym]的点和星行为。从我们的主要定理得出正确的行动是从天然表示诱导\（\ mathcal【W} \）上ħ ⁰（X^小号）。