Summary: COHOMOLOGY OF LINE BUNDLES
HENNING HAAHR ANDERSEN
Let G be a reductive algebraic group over a field k of characteristic
p # 0. If L is a line bundle on the flag variety X for G then the co
homology modules H i (X, L), i # 0 have a natural Gstructure. The
Gmodules arising in this way play a prominent role in the representa
tion theory of G. This is for instance illustrated by the following four
1.1. The Chevalley classification of irreducible Gmodules .
This theorem says that all the finite dimensional irreducible Gmodules
occur as submodules in H 0 (X, L) with L running through the set of
e#ective line bundles on X. Moreover, if p = 0 then any line bundle on
X has at most one nonvanishing cohomology module, and that one is
irreducible. This last result is the BorelWeilBott theorem .
1.2. The strong linkage principle . The BorelWeilBott theorem
mentioned above fails badly when p > 0. As a weaker substitute for this
result we have in positive characteristics the strong linkage principle.
It says that the composition factors of a given cohomology module
H i (X, L), i # 0, L a line bundle on X, have highest weights strongly