Sheaf-Theoretic 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SU(2)和O(3，R)，SU(2)是2x2酉复矩阵的群，O(3，R)是具有实项的3x3正交矩阵的群。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。本研究将利用几何方法在模表示理论方面取得进展。许多预期的结果是由复杂表示理论中的已知事实驱动的，但必须在模块化情况下开发新的工具和技术。在这项研究中，P.I.还将开展针对博士生和其他职业生涯早期研究人员的研究培训活动。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”。这项研究将在这些发展的基础上，以三个不同主题的项目为基础：(I)全球舒伯特变种的拓扑；(Ii)Kazhdan-Lusztig胞元、张量理想和倾斜模数；以及(Iii)相干层的“淤积”复合体。主题(I)与数论和潜在的“模分枝的萨塔克等价”有关。题目(Ii)与表征理论中的经典问题有最直接的联系，而题目(Iii)有望在K-理论和分类方面带来新的研究途径，例如在对称空间的背景下。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 SU(2), the group of 2x2 unitary complex matrices, and O(3,R), the group of orthogonal 3x3 matrices with real entries. 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. This research will use geometric methods to make advances in modular representation theory. Many of the anticipated results are motivated by known facts in complex representation theory, but new tools and techniques must be developed in the modular case. In connection with this research, the P.I. will also undertake research-training activities aimed at Ph.D. students and other early-career researchers. 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 research will build on these developments with projects on three different topics: (i) the topology of global Schubert varieties; (ii) Kazhdan-Lusztig cells, tensor ideals, and tilting modules; and (iii) "silting" complexes of coherent sheaves. Topic (i) has connections to number theory and to a potential "modular ramified Satake equivalence". Topic (ii) has the most direct links to classical questions in representation theory, while topic (iii) is expected to lead to new avenues of research in K-theory and categorification, for instance in the context of symmetric spaces.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.