The cohomology of line bundles on the three-dimensional flag variety

We give a recursive description of the characters of the cohomology of the line bundles of the three-dimensional flag variety over an algebraically closed field of characteristic p. The recursion involves also certain rank two bundles and we calculate their cohomology at the same time. The method of proof is to adapt an expansion formula valid generally for homogeneous vector bundles on flag varieties G/B to the case in which G is the special linear group of degree 3. In order to carry this out it is necessary to calculate explicitly the module of invariants for the action of the first infinitesimal subgroup of the unipotent radical of a Borel subgroup of G on certain tilting modules.