We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.

中文翻译：

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Springer 纤维上同调和 Kostka 多项式的过滤

我们通过 Lusztig 建议的公式证明了一个猜想，该猜想根据广义（单变量）Kostka 多项式表示半单李代数的幂零锥的二阶 Poisson-de Rham 同调。这使我们能够在标志变体上同调上构建一个规范的过滤族，因此在 Weyl 群的不可约表示上，其希尔伯特级数由广义 Kostka 多项式给出。我们推断出所有 Springer 纤维的上同调的结果。特别是，这计算了所有经典有限 W 代数的第零泊松同调的分级，以及所有量子有限 W 代数的第 0 次 Hochschild 同调的过滤，并且我们推广到所有同调度。因此，我们推导出 Proudfoot 关于辛对偶的猜想，在 A 型中，Slodowy 切片的泊松同调与幂零轨道闭合的交叉上同调有关。在最后一节中，我们给出了设置神奇 D 模的主要定理的类比。