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Modular Deligne-Lusztig theory

模块化德利涅-鲁斯蒂格理论

基本信息

批准号：
EP/H026568/1
负责人：
Olivier Dudas
金额：
$ 30.29万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH026568%2F1
关键词：
Modular Deligne Lusztig theory

项目摘要

I study the representation theory of a certain class of finite groups : the finite reductive groups. Since the work of Steinberg, one knows how to construct such a group as the (possibly twisted) rational points of an algebraic reductive group defined over the algebraic closure of a finite field. For example, the group GL(n,q) is obtained by restricting the coefficients of any non-singular matrix to the finite field with q elements. From that geometrical point of view, two directions have been investigated : - how to deduce algebraic and structural properties from the theory of reductive groups over an algebraically closed field; - how to produce representations coming from actions of the finite group on a family of algebraic varieties. This last point has been initiated by Deligne and Lusztig and intensively studied in a series of papers by Lusztig.There are different notions of representations, depending on the coefficient ring. Any representation can be viewed as a lattice or a vector space, together with a linear action of the group. The advantage of the construction of Deligne and Lusztig is that there is a canonical way to linearize the geometric action of the group, with respect to a large family of rings, including the ring of l-adic integers Z_l, where l is a prime number different form the characteristic of the finite field. I am interested in the modular representation theory in non-defining characteristic, that is, the case where this ring is a finite extension of Z_l. In some way, it is the most sophisticated case, since the others are obtained by scalar extensions or reduction mod l. With the pioneer work of Bonnaf and Rouquier in this direction, there are now good evidences that the Deligne-Lusztig varieties should encode many parts of this theory, including blocks, projective modules, decomposition matrices. The main aspect of my work during this Fellowship will be to extract and decode this information. I will be particularly interested in Brou's conjecture and in the modular analogue of Lusztig's character sheaves.

我研究了一类有限群的表示论：有限约化群。由于斯坦伯格的工作，人们知道如何构造这样一个群作为（可能扭曲的）有限域的代数闭包上定义的代数约化群的有理点。例如，群GL（n，q）是通过将任何非奇异矩阵的系数限制在具有q个元素的有限域上而获得的。从几何的角度来看，两个方向进行了调查：-如何推导出代数和结构性质的理论还原群在代数封闭领域; -如何产生陈述来自行动的有限群对家庭的代数簇。这最后一点已开始由德利涅和Lusztig和深入研究的一系列文件的Lusztig。有不同的概念表示，这取决于系数环。任何表示都可以看作是一个格或向量空间，以及群的线性作用。Deligne和Lusztig构造的优点是，对于包括l-adic整数环Z_l在内的大环族，有一个规范的方法来线性化群的几何作用，其中l是不同于有限域特征的素数。本文主要研究非定义特征的模表示理论，即环是Z_l的有限扩张的情形。在某种程度上，它是最复杂的情况，因为其他的是通过标量扩展或约简模1获得的。随着Bonnaf和Rouquier在这个方向上的开创性工作，现在有很好的证据表明Deligne-Lusztig变种应该编码这个理论的许多部分，包括块，投射模，分解矩阵。在这个奖学金期间，我工作的主要方面将是提取和解码这些信息。我将特别感兴趣的是布鲁的猜想，并在模块模拟Lusztig的字符层。