Representation Theory via Cohomology of Line Bundles

Abstract: Let G be a reductive algebraic group over a field k and let B be a Borel subgroup in G. We demonstrate how a number of results on the cohomology of line bundles on the flag manifold G/B have had interesting consequences in the representation theory for G. And vice versa. Our focus is on the case where the characteristic of k is positive. In this case both the vanishing behavior of the cohomology modules for a line bundle on G/B and the G-structures of the non-zero cohomology modules are still very much open problems. We give an account of the developments over the years, trying to illustrate what is now known and what is still not known today.