Cohomology of line bundles on the flag variety $SL_3/B$ can be computed using the Bott-Borel-Weil formula in the case the ground field has characteristic zero. In this way one obtains an explicit formula for the dimensions of the cohomology groups.
Is there a similar formula in the positive characteristic case?
If there is even a generalization to general flag varieties $G/P$, I'd be interested in hearing about that too.