Character sheaves, V
Character sheaves, V
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DOI：
10.1016/0001-8708(86)90071-x
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发表时间：
1985-09
影响因子：
1.7
通讯作者：
G. Lusztig
中科院分区：
文献类型：
--
作者：
G. Lusztig
本文是系列文章 [S, 13, 17, 24] 的一部分，致力于研究代数闭域 k 上的连通还原代数群 G 上的 G 类不可约反常滑轮（称为字符滑轮）。（章节、小节和参考文献的编号将继续前面部分的编号。）本文中的大多数结果都对 k 的特征有非常轻微的限制，请参见 (23.0.1)。 1）。为简单起见，在本介绍中，我们假设 k 的特性对 G 有利；这特别意味着 (23.0.1) 成立。我们的主要结果之一是定理 23.1，它给出了 G 的特征轮的分类，中心的分量组忠实地作用于这些特征轮；此外，它给出了一个与[6]中的主要定理（4.23）非常相似的重数公式；它还指出 G 是干净的（在（13.9.2）的意义上），它满足奇偶条件（15.13）并且 G 上的特征滑轮类别与[4]中定义的允许复合体类别一致。对于 A 类群体和特殊群体，这基本上已在第四部分中完成[24]；本文讨论了经典群的情况（第 22 和 23 节）。我们的结果的应用之一是计算 G 中任何单能类 C 的闭包 C 的局部交集上同调滑轮%'“IC (C, b)，其系数在 C 上的任何 G 等变不可约局部系统 d 中。对于 G= GL,(k)，这是在 [22] 中完成的；对于其他简单的 G，对于那些假设的 (C, B)，它已经在 [27,28,20] 中完成 以施普林格的信件为例[7]。在本文中，我们通过删除 (C, 8) 上的最后一个假设来完成此计算。（参见定理 24.8。）该计算本质上使用了特征滑轮理论。在第 25 节中，表明在 G 在 F 上定义的情况下，特征滑轮 A 的特征函数 1 A、BA （参见（252.1））本身在 Fq 上定义， 形成 G (F,) 上类函数空间的正交基。可以推测，这就是
This paper is part of a series [S, 13, 17, 24] devoted to the study of a class G of irreducible perverse sheaves (called character sheaves) on a connected reductive algebraic group G over an algebraically closed field k.(The numbering of sections, subsections, and references will continue that of the earlier parts.)Most results in this paper hold under a very mild restriction on the characteristic of k, see (23.0. 1). For simplicity, in this introduction, we assume that the characteristic of k is good for G; this implies in particular that (23.0. 1) holds. One of our main results is Theorem 23.1 which gives a classification of the character sheaves of G on which the group of components of the centre acts faithfully; moreover, it gives a multiplicity formula rather analogous to the main theorem (4.23) in [6]; it also states that G is clean (in the sense of (13.9. 2)), it satisfies the parity condition (15.13) and that the class of character sheaves on G coincides with the class of admissible complexes defined in [4]. In the case of groups of type A and exceptional groups, this has been essentially done in part IV [24]; the case of classical groups is dealt with in this paper (Sections 22 and 23). One of the applications of our results is the computation of the local intersection cohomology sheaves%‘“IC (C, b) of the closure C of any unipotent class C in G with coefficients in any G-equivariant irreducible local system d on C. For G= GL,(k), this was done in [22]; for the other simple G, it has been done in [27, 28, 20] for those (C, B) which are assumed to be in the image of Springer’s correspondence [7]. In this paper, we complete this computation by removing the last assumption on (C, 8).(See Theorem 24.8.) The computation uses in an essential way the theory of character sheaves. In Section 25, it is shown that in the case where G is defined over F,, the characteristic functions 1 A, BA (see (252.1)) of the character sheaves A which are themselves defined over Fq, form an orthonormal basis of the space of class functions on G (F,). It may be conjectured that this is the