Mathematical Sciences: Cohomology and Modular Representations of Algebraic Groups

该项目涉及的几个方面， 域上半单代数群的表示理论 的素数特征，并应用于有限群， 说谎的类型。 最近的想法Lusztig有关量子群和 模块化表示将探讨，重点是 内射模 由上同调引起的表示 旗叶品种的系丛性状将是另一个中心问题; 这里的一个主要问题涉及与 仿射Weyl群的Kazhdan-Lusztig多项式。 研究 代数群的性质应该提供对 相关有限单群的表示;这里是一个数 关于内射模的开放性问题和解答， 普通字符的分解行为mod p将是 研究了 这项研究是在一般领域的代数群。 拟议的研究福尔斯两个大标题：上同调 旗簇上的线丛，以及内射模。 沿着这些路线的进展都将揭示卢斯蒂格的 ”“行，行。

This project is concerned with several aspects of the representation theory of semisimple algebraic groups over fields of prime characteristic, with applications to finite groups of Lie type. Recent ideas of Lusztig relating quantum groups and modular representations will be explored, with emphasis on injective modules. Representations arising from the cohomology of line bundles on flag varieties will be another central issue; a major question here concerns the relationship with Kazhdan-Lusztig polynomials for the affine Weyl group. The study of algebraic groups should provide further insight into representations of associated finite simple groups; here a number of open questions and conjectures about injective modules and decomposition behavior mod p of ordinary characters will be studied. This research is in the general area of algebraic groups. The proposed research falls under two broad headings: cohomology of line bundles on flag varieties, and injective modules. Advances along these lines would both shed light on Lusztig's conjecture, as well as point the way beyond it.