Mathematical Sciences: Cohomology and Modular Representations of Algebriac Groups

该奖项支持代数群方面的工作。主要研究人员将研究素特征域上的半单代数群的表示理论的几个方面，并将其应用于李型有限群。Lusztig关于量子群和模表示的程序是主要关注的，重点是内射模和倾斜模。由旗种上的线球的上同调所产生的表示提出了相关的问题；这里的一个主要问题是关于仿射Weyl群与Kazhdon-Luztig多项式的关系。对代数群的研究应该提供对相关的有限单群的表示以及相关上同调的进一步的洞察。这项研究是在代数群的一般领域进行的。粗略地说，满足多项式恒等式的群是代数群。建议的研究分为两个大标题：旗簇上的线丛的上同调和内射模。沿着这些思路取得的进展既可以解释卢兹蒂希的猜想，也可以指出超越猜想的方法。

This award supports work on algebraic groups. The principal investigator will study several aspects of the representation theory of semisimple algebraic groups over fields of prime characteristic, with applications to finite groups of Lie type. The program of Lusztig relating quantum groups and modular representations is of primary concern, with emphasis on injective modules and tilting modules. Representation arising from the cohomology of line budnles on flag varieties pose related problems; a major question here concerns the relationship with Kazhdon-Luztig polynomials for the affine Weyl group. The study of algebraic groups should provide further insight into representations of associated finite simple groups, and related cohomology. This research is in the general area of algebraic groups. Roughly speaking, a group which satisfies a polynomial identity is an algebraic group. 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.