Modular Deligne-Lusztig theory

本文研究一类有限群的表示理论：有限约化群。由于Steinberg的工作，人们知道如何构造这样一个群，即在有限域的代数闭包上定义的代数约化群的（可能是扭曲的）有理点。例如，将任意非奇异矩阵的系数限制在有q个元素的有限域上，得到群GL（n,q）。从几何角度出发，研究了两个方向：如何从代数闭域上的约化群理论推导代数和结构性质；-如何从有限群对代数变量族的作用中产生表示。最后一点是由Deligne和Lusztig提出的，并在Lusztig的一系列论文中进行了深入研究。有不同的表示概念，取决于系数环。任何表示都可以看作是晶格空间或向量空间，以及群的线性作用。Deligne和Lusztig构造的优点是，对于一大族环，包括l进整数环Z_l，其中l是不同于有限域特征的素数，有一种规范的方法来线性化群的几何作用。我感兴趣的是非定义特征的模表示理论，即这个环是Z_l的有限扩展的情况。在某种程度上，它是最复杂的情况，因为其他情况都是通过标量扩展或约简模1得到的。随着Bonnaf和Rouquier在这个方向上的先驱工作，现在有很好的证据表明，delignee - lusztig变体应该编码该理论的许多部分，包括块，投影模块，分解矩阵。我在研究期间的主要工作将是提取和解码这些信息。我会特别感兴趣的是布鲁的猜想和吕兹蒂格的特性的模模拟。

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.