Cohomology of line bundles on flag varieties in positive characteristic

Let G be a semi-simple algebraic group over an algebraically closed field k of positive characteristic and let B be a Borel subgroup. The cohomology of line bundles on the flag variety G/B induced by characters of B are important objects in the representation theory of G. In this talk, I will start by briefly recalling the theory in zero characteristic, which has been well-understood for a long time, and some important previous results in positive characteristics, due to Kempf, Griffith, Andersen, Jantzen, Kuhne-Hausmann, Irving, Doty, Sullivan, Donkin, etc. Then, I will present the new results for G=SL3 obtained in my thesis. More precisely, I have shown the existence of two filtrations of Hi(G/B,μ). The first exists for i=1.2 and μ in the Griffith region. The second, which generalizes the p-filtration introduced by Jantzen, exists for all i and μ.