The cohomology of line bundles on the three dimensional flag variety

The purpose of this paper is to give a recursive description of the characters of the cohomology of the line bundles on the three dimensional flag variety over an algebraically closed field k of characteristic p > 0. In fact our recursive procedure also involves certain rank 2 bundles and we determine the characters of the cohomology of these bundles at the same time. The paper may be regarded as a substantial worked example of the expansion formula for the character of the cohomology of homogeneous vector bundles, on a generalized flag variety G/B (where G is a reductive group over k and B is a Borel subgroup), given in [7]. In Section 1 we set up notation and express the main result of [7] in the form most suitable for the application, to the case G = SL3(k), given here. Section 2 is the heart of the work and we here work out the precise module structure of the module of invariants of certain tilting modules, under the action of the first infinitesimal subgroup of a maximal unipotent subgroup of G, as well as the invariants of the tensor product of tilting modules and certain two dimensional B-modules. In section 3 we give character formulas for certain homogeneous line bundles and rank 2 bundles on G/B determined by “small” weights. These are the base cases for our recursive procedure for finding the characters of the cohomolgy of all line bundles, and certain rank two bundles on G/B. In Section 4 we apply the results of Section 2 to obtain recursive formulas for the characters of cohomology modules in case p = 2. We do this for p = 3 in Section 5 and for p ≥ 5 in Section 6. In Section 7, we give another smaller application of the main formula of [7]. We consider (for any reductive group) the first cohomology group RIndBkμ of the line bundle determined by μ = −pα, where α is a simple root. It is shown in [12;II,5.18 Corollary] that this group has simple socle k for μ = −pα. We show that in fact RIndBkμ is equal to k, if α is not isolated. Our route taken to this result gives us an opportunity to add to the number of proofs of Kempf’s Vanishing Theorem