Informal Friday Seminar: Liu -- Introduction to cohomology of line bundles on flag varieties

Part I: The cohomology of line bundles on flag varieties is a natural and important
object in the representation theory of reductive algebraic groups.  In the first
lecture, I will give an introduction to this topic from the very beginning and try to
give concrete examples of every abstract general definition, so the only prerequisite is
linear algebra.  If time allows, I will also (pretend to) prove some famous fundamental
theorems, such as the classification of all simple G-modules by highest weights, Kempf’s
vanishing theorem, Borel-Weil-Bott theorem, etc..  I will explain why everything is neat
and nice in the world of characteristic 0 and the difficulties we encounter in positive
characteristic.  

Part II: During last week’s lecture, we’ve talked about the general theory and I
explained why the proof of Borel-Weil-Bott theorem only applies to the characteristic 0
case.  This week I will start from introducing several methods in positive
characteristic theory and I will talk about some important previous results in this
problem.  Then I will talk about the new results obtained in my thesis, where I proved
the existence of two filtrations of the cohomology of line bundles on the three-
dimensional flag variety (corresponding to G=SL_3), of which the second one generalizes
Jantzen’s p-filtration.  In particular, as a corollary, I will recover the recursive
formulae of the characters proved by Donkin in 2006.