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.
