Loewy Series of Certain Indecomposable Modules for Frobenius Subgroups

We imitate some approaches in infinite dimensional representation theory of complex semisimple Lie algebras by using the truncated category method in the categories of modules for certain Frobenius subgroups of a semisimple algebraic group over an algebraically closed field of characteristic p > 0. By studying the translation functors from p-singular weights to p-regular weights, we obtain some results on Loewy series of certain indecomposable modules. Introduction Lusztig’s conjecture in the modular representation theory states that the character of a simple module can be calculated from a set of polynomials related with the affine Weyl group. Andersen’s work [1] shows that Lusztig’s conjecture is actually equivalent to the following fact: when one translates the simple module across an upper alcove wall, one will get a module which has Loewy length at most three. So the study of the translation functors turns out to be interesting and important. However, the behavior of translation functors is far from being understood. In this paper, we study the translation functors from p-singular weights to p-regular weights to obtain some results on the Loewy series of certain modules for Frobenius subgroups. The representation theory of complex semisimple Lie algebras is an important source of methods in the representation theory of algebraic groups over fields of prime characteristic. These two theories are somehow parallel. The Lusztig conjecture is analogous to the Kazhdan-Lusztig conjecture and the Jantzen conjecture is analogous to the Jantzen conjecture for semisimple Lie algebras. 1980 Mathematics Subject Classification (1985 Revision) Primary 20G05, secondary 18G05