B-cohomology

Let B be a Borel subgroup in a reductive algebraic group G over a field k. We study the cohomology H•(B,λ) of 1-dimensional B-modules λ. When image there is an easy and well-known description of this cohomology whereas the corresponding problem in characteristic p>0 is wide open. We develop some new techniques which enable us to calculate all such cohomology in degrees at most 3 when p is larger than the Coxeter number for G. Our methods also apply to the corresponding question for quantum groups at roots of unity.