Sheaf-Theoretic Methods in Modular Representation Theory

矩阵群是一组可逆方阵，包含其成员的所有乘积和逆矩阵。 典型的例子包括SU（2），2x2酉复矩阵群，和O（3，R），具有真实的元素的正交3x 3矩阵群。 广义地说，表示论的主题是研究这样的群如何通过线性变换作用于复向量空间。 然后，人们会问，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。 模表示论关注的是在这样的域中有元素的矩阵群，作用于同一域上的向量空间。 本研究将使用几何方法来推进模表示理论。许多预期的结果是由复杂表示理论中的已知事实所激发的，但是在模块化的情况下必须开发新的工具和技术。 在这项研究中，P.I.还将开展针对博士的研究培训活动。学生和其他早期职业研究人员。 在过去的几年里，出现了一些强有力的新工具，将几何方法应用于正特征代数群的表示论，包括“宇称层”和“混合模导出范畴”。“这项研究将建立在这些发展的基础上，项目涉及三个不同的主题：（i）全球舒伯特簇的拓扑结构;（ii）Kazhdan-Lusztig细胞，张量理想和倾斜模;（iii）相干层的“淤积”复合体。题目（i）与数论和一个潜在的“模分歧佐竹等价”有关。 主题（ii）与表征理论中的经典问题有着最直接的联系，而主题（iii）有望为K理论和范畴化的研究开辟新的途径，例如在对称空间的背景下。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.