Characteristic Cycles in Hermitian Symmetric Spaces
Characteristic Cycles in Hermitian Symmetric Spaces
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厄米对称空间中的特征循环
DOI：
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发表时间：
1997
期刊：
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通讯作者：
Joseph H. G. Fu
中科院分区：
文献类型：
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作者：
Brian D. Boe;Joseph H. G. Fu
本文给出了经典Hermitian对称空间中Schubert簇X上某些典型层的特征圈的显式组合表达式：即交同调层IHX和常数层IHX。三种主要的有趣的情形是An型群（标准格拉斯曼群）、Cn型群（拉格朗日格拉斯曼群）和Dn型群的厄米特对称空间。特别地，我们发现CC（IH X）对所有Schubert簇X是不可约的当且仅当相关的Dynkin簇是简单的laced.对于标准格拉斯曼的舒伯特簇的结果已经由Bressler、Finkelberg和Lunts在早期建立，而Cn和Dn情形的计算是新的。我们的方法是先用直接几何方法计算CC（IHX），然后用Kazhdan-Lusztig多项式的组合数学方法（简化为Hermitian对称空间）计算CC（IHX）.几何方法基于基本公式，其中Xr ↓ X构成围绕变量X的管族。这个公式立刻导出了CC（λ X）的系数的表达式，它是球面之间某些奇异映射的度数。
Abstract We give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties X in the classical Hermitian symmetric spaces: namely the intersection homology sheaves IH X and the constant sheaves ℂ X . The three main cases of interest are the Hermitian symmetric spaces for groups of type An (the standard Grassmannian), Cn (the Lagrangian Grassmannian) and Dn . In particular we find that CC(IH X ) is irreducible for all Schubert varieties X if and only if the associated Dynkin diagramis simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the Cn and Dn cases are new. Our approach is to compute CC(ℂ X ) by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute CC(IH X ). The geometric method is based on the fundamental formula where the Xr ↓ X constitute a family of tubes around the variety X. This formula leads at once to an expression for the coefficients of CC(ℂ X ) as the degrees of certain singular maps between spheres.