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。 典型示例包括 SU(2)（2x2 酉复矩阵群）和 O(3,R)（具有实数项的正交 3x3 矩阵群）。 从广义上讲，表示论的主题涉及这些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）替换复数，会发生什么。 模块化表示理论涉及具有此类域中的条目的矩阵组，作用于同一域上的向量空间。 本研究将利用几何方法来推进模表示理论。许多预期结果都是由复杂表示理论中的已知事实激发的，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.还将开展针对博士学位的研究培训活动。学生和其他早期职业研究人员。 过去几年出现了将几何方法应用于正特性代数群表示论的强大新工具，包括“奇偶滑轮”和“混合模派生范畴”。这项研究将建立在这些进展的基础上，通过三个不同主题的项目：（i）全球舒伯特变体的拓扑结构； (ii) Kazhdan-Lusztig 单元、张量理想和倾斜模块； (iii) 连贯滑轮的“淤积”复合体。主题 (i) 与数论和潜在的“模分支佐竹等价”有关。 主题 (ii) 与表示论中的经典问题有最直接的联系，而主题 (iii) 预计将带来 K 理论和分类的新研究途径，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 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.