Reductive pairs arising from representations

TL;DR

This paper investigates when the pair of groups formed by a reductive algebraic group and its representation is itself reductive, providing complete classifications for SL2 in positive characteristic and examples for SL3.

Contribution

It offers a comprehensive analysis of reductive pairs arising from representations, especially characterizing such pairs for SL2 in positive characteristic based on highest weight expansions.

Findings

Complete classification for SL2(K) in positive characteristic

Behavior determined by base p expansion of highest weight

Examples illustrating the theoretical results for SL3(K)

Abstract

Let G be a reductive algebraic group and V a G-module. We consider the question of when (GL(V), rho(G)) is a reductive pair of algebraic groups, where rho is the representation afforded by V. We first make some observations about general G and V, then specialise to the group SL2(K) with K algebraically closed of positive characteristic p. For this group we provide complete answers for the classes of simple and Weyl modules, the behaviour being determined by the base p expansion of the highest weight of the module. We conclude by illustrating some of the results from the first section with examples for the group SL3(K).