GREEN POLYNOMIALS AND SINGULARITIES OF UNIPOTENT CLASSES
GREEN POLYNOMIALS AND SINGULARITIES OF UNIPOTENT CLASSES
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DOI：
10.1016/0001-8708(81)90038-4
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发表时间：
1981-01-01
影响因子：
1.7
通讯作者：
LUSZTIG, G
中科院分区：
文献类型：
--
作者：
LUSZTIG, G
Deligne [2]将一个I进层复形“Q”与X相联系（规范直至准同构），具有可构造的上同调层F '（X），在导出范畴中是自对偶的，等价于在X的光滑部分上以0度化为常数层Q的复形，并且具有性质：对于i< 0，Z@（X）= 0，如果i> 0，则p（X）具有维度Q d-i-1的支持。他的建设，这是勾画在[8，第31节是一个代数类似的Goresky和麦克弗森中交上同调理论[3，4]。我们称Z '（X）为X的DGM层。本文的目的是描述这一理论在有限群GL，（iF，）的不可约特征标研究中的一个应用。设k是IF的一个代数闭包。令A=（Ai> I，>，. 2 1，（2 0））是n的一个分拆：n= A，+ A，+*+ An.我们将幂幺类X，c GL，（k）与1联系起来，它由幂幺元素组成，这些幂幺元素的Jordan块的大小为A，我，。我们还将GL，（1F，）的不可约幂幺表示E，与1相关联：它是由维数为Ai，Ai + Ai，Ai + Ai + As，.的子空间的标志的稳定子的单位表示所诱导的表示的“最大”分量，在F中：考虑X的闭包的DGM层F”（xA）。在下面的定理中，层@（fA）将被认为是GL，（k）中幂单元的整个簇上的层，在XL的补上等于零。
Deligne [2] has associated to X a complex “Q, of I-adic sheaves (canonical up to quasi-isomorphism) which has constructible cohomology sheaves, F’(X), which is self-dual in the derived category, which is equivalent to the complex reduced to constant sheaf Q, in degree 0 over the smooth part of X, and which has the property: Z@(X)= 0 for i< 0, p (X) has support of dimension Q d-i-1 if i> 0. His construction, which is sketched in [8, Sect. 31 is an algebraic analogue of the Goresky and Macpherson middle intersection cohomology theory [3, 4]. We shall call Z’(X) the DGM sheaves of X. The purpose of this paper is to describe an application of this theory to the study of irreducible characters of the finite group GL,(iF,). Let k be an algebraic closure of IF,. Let A=(Ai> I,>,... 2 1,(2 0)) be a partition of n: n= A,+ A,+***+ An. We associate to 1 the unipotent class X, c GL,(k) consisting of the unipotent elements which have Jordan blocks of size A,, &,..., I,. We also associate to 1 the irreducible unipotent representation E, of GL,(lF,): it is the “biggest” component of the representation induced by the identity representation of the stabilizer of a flag of subspaces of dimensions A,, A,+ A,, Ai+ A,+ As,..., in F:. Consider the DGM sheaves, F”(xA) of the closure of X,. In the following theorem the sheaves@(fA) will be regarded as sheaves on the whole variety of unipotent elements in GL,(k), equal to zero on the complement of XL.