G-modules, Springer's representations and bivariant Chern classes
G-modules, Springer's representations and bivariant Chern classes
复制标题
G 模块、Springer 表示和二变 Chern 类
DOI：
10.1016/0001-8708(86)90064-2
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发表时间：
1986
影响因子：
1.7
通讯作者：
V. Ginsburg
中科院分区：
文献类型：
--
作者：
V. Ginsburg
0.1.设G是连通单连通复半单李群，其李代数为6，Bore 1子群为B.基本上有两种不同的方法来研究简单G-模。第一个与Flag流形X= G/B有关，G-模与X上的向量丛和层有关。第二种方法，所谓的Kirillov-Kostant“轨道方法”，将表示与李代数8的对偶空间6* 中的余伴随轨道联系起来。1981年，Borho和Brylinski以及作者独立地提出了将这两个图像连接在一起的想法，从而用6* x X中的特征变种来描述B-模和本原理想。在最近的一篇论文中，Kashiwara和Tanisaki [KT]将Flag流形上完整系统的特征循环与Weyl群表示联系起来。在这里，我们解释这种关系的一个新的“拉格朗日”建设的施普林格的陈述。我们还表明，施普林格的陈述反过来只是一个特殊的情况下，一般双变理论的陈类奇异品种。关于B-模，我们给出复半单李代数的包络代数中本原理想的伴随簇不可约的两个证明。他们中的第一个（见第1节）出现在[KT]，并在很大程度上基于约瑟夫的结果。第二（见第8节），而独立的约瑟夫的结果，。使用Kazhdan-Lusztig细胞的某些性质，由Barbasch和Vogan和Lusztig验证。1
0.1. Let G be a connected simply connected complex semi-simple Lie group with Lie algebra 6 and Bore1 subgroup B. There are basically two different approaches to the study of simple G-modules. The first one is connected with the Flag manifold X= G/B and G-modules are related to vector bundles and sheaves on X. The second approach, the so-called Kirillov-Kostant“orbit method,” links representations with coadjoint orbits in the dual space 6* of the Lie algebra 8. In 1981 Borho and Brylinski and the author independently developed the idea of joining these two pictures together, thus describing B-modules and primitive ideals by their characteristic varieties in 6* x X. In a recent paper Kashiwara and Tanisaki [KT] related characteristic cycles of holonomic systems on the Flag manifold to Weyl group representations. Here we explain that relation by means of a new “Lagrangian” construction of Springer’s representations. We also show that Springer’s representations are in turn just a special case of the general bivariant theory of Chern classes for singular varieties. With regard to B-modules we will give two proofs of the irreducibility of the associated variety of a primitive ideal in an enveloping algebra of a complex semi-simple Lie algebra. The first of them (see Section 1) appeared in [KT] and is heavily based on results of Joseph. The second (see Section 8), while independent of Joseph’s results,. uses a certain property of Kazhdan-Lusztig cells, verified by Barbasch and Vogan and Lusztig. 1