Geometric Methods in Modular Representation Theory

矩阵群是一组可逆的方阵，它包含了矩阵中所有元素的乘积和逆矩阵。典型的例子有O(3，R)， 3x3正交实数矩阵群，SU(2)， 2x2酉复矩阵群。广义地说，表示理论的主题是处理这样的群如何通过线性变换作用于（复）向量空间。然后有人会问，如果我们用有限域（或有限域的代数闭包）代替复数会发生什么。模表示理论关注的是在这样一个域中有元素的矩阵群，它们作用于同一域上的向量空间。本文的研究将推动几何方法的模表示理论的发展。许多预期的结果类似于复杂表示理论中的已知事实，但是在模块化的情况下必须开发新的工具和技术。在过去的几年中，出现了一些强大的新工具，将几何方法应用于代数群的正特征表示理论，包括宇称轴和混合模派生范畴。本项目将以这些发展为基础，开展三个不同主题的项目：(i)正特征的单态算子；（ii）相干束的广义施普林格理论；（iii） Kazhdan-Lusztig单元、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）有望对表征理论产生具体的影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.