Geometric Methods in Modular Representation Theory

矩阵群是一组包含其成员的所有乘积和逆的可逆方阵。典型的例子包括O(3，R)和SU(2)，O(3，R)是具有实项的3x3正交矩阵群，SU(2)是2x2酉复矩阵群。广义地讲，表示论的主题是这样的群如何通过线性变换作用于(复杂的)向量空间。然后，人们可以问，如果我们用有限域(或有限域的代数闭包)来代替复数，会发生什么。模表示理论涉及这样一个域中具有条目的矩阵群，作用于同一域上的向量空间。本文的研究将使几何方法在模表示理论方面取得新的进展。许多预期的结果与复数表示理论中的已知事实相似，但必须在模块化情况下开发新的工具和技术。在过去的几年里，出现了将几何方法应用于正特征代数群表示理论的强大的新工具，包括奇偶层和混合模导出范畴。这个项目将在这些发展的基础上，在三个不同的主题上进行项目：(I)正特征的单行算子；(Ii)相干层的广义Springer理论；以及(Iii)Kazhdan-Lusztig胞格、张量理想和倾斜模。主题(I)本质上是几何性质的，而主题(Ii)和(Iii)预计将对代表理论产生具体的影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A matrix group is a set of invertible square matrices that contains all products and inverses of its members. Typical examples include O(3,R), the group of orthogonal 3x3 matrices with real entries, and SU(2), the group of 2x2 unitary complex matrices. Broadly speaking, the subject of representation theory deals with how such groups can act on a (complex) vector space via linear transformations. One can then ask what happens if we replace the complex numbers by a finite field (or the algebraic closure of a finite field). Modular representation theory is concerned with matrix groups with entries in such a field, acting on vector spaces over the same field. The proposed research will make advances in modular representation theory by geometric methods. Many of the anticipated results are analogous to known facts in complex representation theory, but new tools and techniques must be developed in the modular case. The past few years have seen the emergence of powerful new tools for applying geometric methods to the representation theory of algebraic groups in positive characteristic, including parity sheaves and the mixed modular derived category. This project will build on these developments with projects on three different topics: (i) monodromy operators in positive characteristic; (ii) generalized Springer theory for coherent sheaves; and (iii) Kazhdan-Lusztig cells, tensor ideals, and tilting modules. Topic (i) is essentially geometric in nature, while topics (ii) and (iii) are expected to have concrete consequences for representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.