On the cohomology of certain homogeneous vector bundles of G/B in characteristic zero

In his famous paper Demazure (1976) [2], Demazure gave a very short proof of the Borel–Weil–Bott theorem, giving the cohomology of line bundles on a generalized flag variety over a field of characteristic zero. To do this Demazure worked with certain natural modules denoted by Vλ,α and their associated vector bundles. He then used dimension shifting and the modules Vλ,α to explicitly derive the cohomology of line bundles on a generalized flag variety. He did this without describing the cohomology of the vector bundles corresponding to these auxiliary modules. However we believe the homological properties of Vλ,α to be of independent interest. Here we calculate the cohomology of vector bundles corresponding to these modules and some generalizations.