Some Good-Filtration Subgroups of Simple Algebraic Groups

TL;DR

This paper investigates when certain subgroups of classical and exceptional algebraic groups preserve the property of having a good filtration in their modules, identifying specific cases and primes where this holds.

Contribution

It characterizes optimal SL2-subgroups and subsystem subgroups as good filtration subgroups in classical and exceptional groups, specifying prime conditions.

Findings

Optimal SL2-subgroups are good filtration subgroups in classical groups.

Identifies primes for which exceptional groups' subgroups are good filtration subgroups.

Determines conditions for subsystem subgroups to be good filtration subgroups.

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.