BB-cohomology

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