Algebraic groups in characteristic p: line bundles, Ext groups and sum formulas
Dr. Upendra Kulkarni
I will sketch some recent work in the representation theory of reductive algebraic groups in positive characteristic. An important role is played by Jantzen's filtration of Weyl modules.
More recently a similar-looking filtration involving tilting modules was defined by H.H. Andersen. Both these filtrations obey similar sum formulas, though in the tilting case the formula was not known in full generality.
After introducing the necessary objects, I will show how both sum formulas can be understood as calculations of the same Euler characteristic. For this it is essential to work with the corresponding Chevalley group over the integers. I will then indicate how to give a uniform and complete proof of both sum formulas simultaneously via the homological link. The proof uses cohomology of line bundles over the flag variety. I will sketch the classical Borel-Weil-Bott theory and what happens to it over the integers.
The talk is based on joint work with H.H. Andersen.