Virtual Workshop on Recent Developments in Geometric Representation Theory

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 $G = SL_3$ obtained in my thesis. More precisely, I have shown the existence of two filtrations of $H^i (G/B, \mu)$. The first exists for $i = 1.2$ and $\mu$ in the Griffith region. The second, which generalizes the $p$-filtration introduced by Jantzen, exists for all $i$ and $\mu$.