Mathematical Sciences: Cohomology and Modular Representations of Algebraic Groups

本文研究了素数特征域上半简单代数群的表示理论的几个方面，并在李型有限群中的应用。本课程将探讨有关量子群和模表示的Lusztig最新思想，重点是内射模。由旗种上的线束的上同源性引起的表示将是另一个中心问题；这里的一个主要问题涉及仿射Weyl群与Kazhdan-Lusztig多项式的关系。代数群的研究应该提供对相关有限简单群的表示的进一步见解；本文将研究普通字符的内射模和分解行为模p的一些开放性问题和猜想。这项研究是在代数群的一般领域。提出的研究分为两大类：旗种上线束的上同性和注入模。沿着这条路线的进展，既能阐明Lusztig的猜想，也能指出超越它的道路。

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.