Modular Deligne-Lusztig theory

我研究了一类有限群的表示理论：有限约化群。自从Steinberg的工作以来，人们知道如何构造这样一个群，即定义在有限域的代数闭包上的代数约化群的(可能是扭曲的)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得到的。从这个几何观点出发，研究了两个方向：-如何从代数闭域上的约化群理论推导出代数性质和结构性质；-如何从有限群在一族代数簇上的作用产生表示。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的研究。根据系数环的不同，有不同的表示概念。任何表示都可以看作是一个格子或一个向量空间，以及群的线性作用。构造Deligne和Lusztig的优点是有一种规范的方法来线性化群的几何作用，关于一大族环，包括L-进整数环Z_L，其中L是一个不同于有限域特征的素数。我对非定义性的模表示理论感兴趣，也就是说，这个环是Z_L的有限扩张。在某种程度上，这是最复杂的情况，因为其他的都是通过标量扩张或约简模L得到的。随着Bonnaf和Rouquier在这方面的开创性工作，现在有很好的证据表明Deligne-Lusztig簇应该编码这一理论的许多部分，包括块，投射模，分解矩阵。在此期间，我工作的主要方面将是提取和破译这些信息。我将对Brou的猜想和Lusztig的特征线的模类比特别感兴趣。

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.