Type: Seminar

Modified: Wed 9 Oct 2019 1629

Distribution: World

CalTitle1: Liu - Introduction to cohomology of line bundles on flag varieties, part I

Calendar2: 11 Oct 2019 1430-1630

CalLoc2: Quad S225

CalTitle2: Liu - Introduction to cohomology of line bundles on flag varieties, part II

Auth: romanova@10.17.127.170 (arom8272) in SMS-SAML

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 weekâ€™s lecture, weâ€™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 Jantzenâ€™s p-filtration. In particular, as a corollary, I will recover the recursive formulae of the characters proved by Donkin in 2006.