Some Good-Filtration Subgroups of Simple Algebraic Groups

Abstract

Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p > 0. An interesting class of representations of G consists of those G-modules having a good filtration -- i.e. a filtration whose layers are the standard highest weight modules obtained as the space of global sections of G-linearized line bundles on the flag variety of G. Let H be a connected and reductive subgroup of G. One says that (G,H) is a Donkin pair, or that H is a good filtration subgroup of G, if every G-module with good filtration also has a good filtration as an H-module. In this paper, we show when G is a "classical group" that the optimal SL2-subgroups of G are good filtration subgroups, and we also determine some primes for which distinguished optimal SL2-subgroups of groups of exceptional type are good filtration subgroups. Additionally, we consider the cases of subsystem subgroups in all types and determine some primes for which they are good filtration subgroups.