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Irreducible A_1 Subgroups of Exceptional Algebraic Groups

Adam Thomas

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 $G$.

Irreducible A_1 Subgroups of Exceptional Algebraic Groups

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 subgroups of exceptional algebraic groups . Consequences are given concerning the representations of such subgroups on various -modules: for example, the conjugacy classes of irreducible subgroups are determined by their composition factors on the adjoint module of .

Paper Structure

This paper contains 13 sections, 24 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Strategy for the proof of Theorem \ref{['mainthm']}
  5. Proof of Theorem \ref{['thmG2']}: $G_2$-irreducible $A_1$ subgroups
  6. Proof of Theorem \ref{['thmF4']}: $F_4$-irreducible $A_1$ subgroups
  7. Proof of Theorem \ref{['thmE6']}: $E_6$-irreducible $A_1$ subgroups
  8. Proof of Theorem \ref{['thmE7']}: $E_7$-irreducible $A_1$ subgroups
  9. Proof of Theorem \ref{['thmE8']}: $E_8$-irreducible $A_1$ subgroups
  10. Corollaries
  11. Tables
  12. Conditions for conjugacy class representatives of $G$-irreducible $A_1$ subgroups of $G = E_7$ and $E_8$
  13. Levi subgroups

Key Result

Theorem 1

Suppose $X$ is a $G$-irreducible subgroup $A_1$ of a simple exceptional algebraic group $G$. Then $X$ is conjugate to exactly one subgroup of Tables G2tab to E8tab, found in Sections secG2 to secE8, respectively and each subgroup in the tables is $G$-irreducible.

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Theorem 3.1: LS1
  • Lemma 3.2: LT
  • Lemma 3.3: tho1
  • Lemma 3.4: LT
  • ...and 24 more