Summary: COHOMOLOGY OF LINE BUNDLES
HENNING HAAHR ANDERSEN
1. Introduction
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
results.
1.1. The Chevalley classification of irreducible G­modules [15].
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 [13].
1.2. The strong linkage principle [2]. 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

Source: Andersen, Henning Haahr - Department of Mathematics, Aarhus Universitet

Collections: Mathematics