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 induced highest weight modules obtained as the space of global sections of G-linearized line bundles on the flag variety of G. Let H⊂G 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 whenever the G-module V has a good filtration, the H-module resHGV has a good filtration.In this paper, we show when G is a “classical group” that the optimalSL2-subgroups of G are good filtration subgroups. We also consider the cases of subsystem subgroups in all types and determine some primes for which they are good filtration subgroups.