等谱交换簇、Harish-Chandra D-模和主幂零对
Isospectral commuting variety, the Harish-Chandra D-module, and principal nilpotent pairs
摘要：
设g是复约化李代数，其Cartan代数为h. Hotta和Kashiwara定义了g × h上的完整D-模M，称为Harish-Chandra模。我们把gr（M），一个关于M上的一个标准Hodge滤子的相伴的分次模，与等谱交换簇，g × g × h × h的一个子簇联系起来，它是g的交换元对簇的一个分歧覆盖.我们的主要结果建立了gr（M）与等谱交换簇的正规化X_范数的结构层的同构。由于Hodge模的理论，等谱交换簇的正规化是Cohen-Macaulay和Gorenstein，证实了M.海曼利用Saito的Hodge D-模理论，我们证明了X_模是Cohen-Macaulay和Gorenstein模.这证实了M.海曼对于g中的任一主幂零对，都存在一个有限的X_范数子概型.相应的坐标环是一个二阶有限维Gorenstein代数，它提供了外尔群的正则表示。该代数的基柱是由h × h上的显著W-调和多项式生成的1维向量空间。在g=gl_n的特殊情况下，上述代数与n！Haiman定理，我们的W-调和多项式化为Garsia-Haiman多项式。此外，在gl_n的情况下，层gr（M）在C^2中的n个点的希尔伯特方案上产生一个向量丛，它与Procesi丛同构。我们的结果被I.戈登获得了一个新的证明积极的Kostka-Macdonald多项式建立较早的海曼。
英文摘要：
Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We relate gr(M), an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g x g x h x h which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of X_norm, the normalization of the isospectral commuting variety. It follows, thanks to the theory of Hodge modules, that the normalization of the isospectral commuting variety is Cohen-Macaulay and Gorenstein, confirming a conjecture of M. Haiman. We deduce, using Saito's theory of Hodge D-modules, that the scheme X_norm is Cohen-Macaulay and Gorenstein. This confirms a conjecture of M. Haiman. Associated with any principal nilpotent pair in g, there is a finite subscheme of X_norm. The corresponding coordinate ring is a bigraded finite dimensional Gorenstein algebra that affords the regular representation of the Weyl group. The socle of that algebra is a 1-dimensional vector space generated by a remarkable W-harmonic polynomial on h x h. In the special case where g=gl_n the above algebras are closely related to the n!-theorem of Haiman and our W-harmonic polynomial reduces to the Garsia-Haiman polynomial. Furthermore, in the gl_n case, the sheaf gr(M) gives rise to a vector bundle on the Hilbert scheme of n points in C^2 that turns out to be isomorphic to the Procesi bundle. Our results were used by I. Gordon to obtain a new proof of positivity of the Kostka-Macdonald polynomials established earlier by Haiman.
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