Irreducible A_1 Subgroups of Exceptional Algebraic Groups

TL;DR

This paper classifies all irreducible A_1 subgroups within exceptional algebraic groups and explores their representations on various modules, providing a comprehensive understanding of their conjugacy classes and structure.

Contribution

It provides the first complete classification of irreducible A_1 subgroups in exceptional algebraic groups and analyzes their representation theory on key modules.

Findings

Classification of all irreducible A_1 subgroups in exceptional groups

Determination of conjugacy classes via composition factors

Insights into subgroup representations on G-modules

Abstract

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible $A_1$ subgroups of exceptional algebraic groups $G$. Consequences are given concerning the representations of such subgroups on various $G$-modules: for example, the conjugacy classes of irreducible $A_1$ subgroups are determined by their composition factors on the adjoint module of $G$.