Parabolic Group Actions and Fans

Let G be an algebraic group and V a linear representation of G. Then it is
desirable to classify all pairs (G,V) so that G acts with a dense orbit on
V. The representation V is then called prehomogeneous. This problem is of
particular interest for parabolic groups acting on an ideal in the
Lie algebra of the unipotent radical. By a classical result of Richardson
it is known that the unipotent radical is always prehomogeneous for the
adjoint action of the parabolic subgroup.
To solve this problem for a large class of groups we use representation
theory of finite dimensional algebras, in particular so-called
quasi-hereditary algebras. The principal idea is to construct for certain
classes of group actions an additive category (a subcategory of a module
category) so that the space is prehomogeneous precisely when there exists
a corresponding module without selfextensions. Then one only needs to
classify certain modules without selfextensions.
We start to explain the main techniques in some well-known example and
introduce the notion of a fan. In a second part we apply these techniques
to the original problem.