Structure of cohomology of line bundles on [formula omitted] for semisimple groups

Abstract

Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p > 0, T a maximal torus of G and B a Borel subgroup containing T . Each weight in X(T ) determines a line bundle on the flag varietyG/B. It turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of B-modules to the category of G-modules for each 1-dimensional B-module defined by a weight in X(T ). Further, IndGBλ = H (G/B, λ) turns out to be the dual of a Weyl module. One of the main problems is to calculate the characters of the irreducible G-modules. For p = 0, the character of the irreducible G-module of highest weight λ is given by Weyl’s character formula and the G-module structures of the cohomology of line bundles are well understood. However, for p > 0, the story is quite different and many of the results remain conjectural. Since characters of Weyl modules are given by Weyl’s character formula, understanding the structure ofH(G/B, λ) turns out to be the main problem. It is also interesting to understand the structure of the higher cohomology, which might help us to understand H. H. Andersen has a series of papers toward the understanding of the higher cohomology, such as the simple socle of H and filtrations of H i [4]. In [5] , using the representations of infinitesimal subgroup schemes of G, he proved generically that H(G/B,w ·λ) has simple socle and simple head and their highest weights can be calculated. In [6], he proved that the socle series of H(λ) comes generically from the