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.
