The discrete series of GLn over a finite field
The discrete series of GLn over a finite field
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有限域上 GLn 的离散级数
DOI：
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发表时间：
1974
期刊：
影响因子：
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通讯作者：
G. Lusztig
中科院分区：
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作者：
G. Lusztig
在这本书中，Lusztig教授通过全新的方法解决了一个有趣的问题：具体来说，使用建筑物和相关复合体的上同调。这本书给出了一个明确的建设一个杰出的成员，D（V），离散系列的GLn（Fq），其中V是n维F-向量空间上的GLn（Fq）的行为。这是一个p-adic表示;更确切地说，D（V）是秩为（q-1）（q2-1）的自由模...（qn-1-1）在F.在第一章中，作者研究了偏序集的同调，证明了一些与几何结构相关的偏序集的同调消失定理。第二章研究有限域上仿射群的表示。第三章定义了D（V），并确定了它对抛物子群的限制。在第四章中，作者计算了D（V）的特征标，并说明了如何利用D（V）的伽罗瓦自同构来得到离散级数的其他成员。应用程序在第5章。作为其研究的主要应用之一，作者给出了GLn（Fq）的标准表示的Brauer提升的精确分析。
In this book Professor Lusztig solves an interesting problem by entirely new methods: specifically, the use of cohomology of buildings and related complexes. The book gives an explicit construction of one distinguished member, D(V), of the discrete series of GLn (Fq), where V is the n-dimensional F-vector space on which GLn(Fq) acts. This is a p-adic representation; more precisely D(V) is a free module of rank (q--1) (q2-1)...(qn-1-1) over the ring of Witt vectors WF of F. In Chapter 1 the author studies the homology of partially ordered sets, and proves some vanishing theorems for the homology of some partially ordered sets associated to geometric structures. Chapter 2 is a study of the representation of the affine group over a finite field. In Chapter 3 D(V) is defined, and its restriction to parabolic subgroups is determined. In Chapter 4 the author computes the character of D(V), and shows how to obtain other members of the discrete series by applying Galois automorphisms to D(V). Applications are in Chapter 5. As one of the main applications of his study the author gives a precise analysis of a Brauer lifting of the standard representation of GLn(Fq).