Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety

Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G. In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when 〈 λ , α ∨ 〉 = − p n a − 1 , ( 1 ≤ a ≤ p , n > 0 ) or 〈 λ , α ∨ 〉 = − p n − r , ( r ≥ 2 , n ≥ 0 ) . We also give the corresponding results for the two dimensional modules N α ( λ ) . These results will help us understand the representations of S L 3 ( k ) in the given cases