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Automorphisms and opposition in spherical buildings of exceptional type, V: The $\mathsf{E}_8$ case

James Parkinson, Hendrik Van Maldeghem

TL;DR

The paper completes the classification of domestic automorphisms for large spherical buildings of exceptional type by resolving the $E_8$ case. It combines explicit Chevalley-group calculations with a geometric framework built from long root subgroup geometries and equator geometries to identify two main classes of domestic automorphisms, Class I and Class II, corresponding to fixed metasymplectic and equator substructures, respectively. The authors prove that any domestic automorphism fixing no chamber must lie in one of these two classes and derive a Density Theorem for $E_8$-type groups, linking conjugacy classes to a small number of $B$-cosets. They also describe concrete representatives and conditions for existence of such automorphisms, and provide spectral-analytic consequences for displacement spectra, contributing to the broader program of understanding conjugacy and fixed-point phenomena in Chevalley groups of exceptional type.

Abstract

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber to an opposite chamber. In previous work the classification of domestic automorphisms in large spherical buildings of types $\mathsf{F}_4$, $\mathsf{E}_6$, and $\mathsf{E}_7$ have been obtained, and in the present paper we complete the classification of domestic automorphisms of large spherical buildings of exceptional type of rank at least~$3$ by classifying such automorphisms in the $\mathsf{E}_8$ case. Applications of this classification are provided, including Density Theorems showing that each conjugacy class in a group acting strongly transitively on a spherical building intersects a very small number of $B$-cosets, with $B$ the stabiliser of a fixed choice of chamber.

Automorphisms and opposition in spherical buildings of exceptional type, V: The $\mathsf{E}_8$ case

TL;DR

The paper completes the classification of domestic automorphisms for large spherical buildings of exceptional type by resolving the case. It combines explicit Chevalley-group calculations with a geometric framework built from long root subgroup geometries and equator geometries to identify two main classes of domestic automorphisms, Class I and Class II, corresponding to fixed metasymplectic and equator substructures, respectively. The authors prove that any domestic automorphism fixing no chamber must lie in one of these two classes and derive a Density Theorem for -type groups, linking conjugacy classes to a small number of -cosets. They also describe concrete representatives and conditions for existence of such automorphisms, and provide spectral-analytic consequences for displacement spectra, contributing to the broader program of understanding conjugacy and fixed-point phenomena in Chevalley groups of exceptional type.

Abstract

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber to an opposite chamber. In previous work the classification of domestic automorphisms in large spherical buildings of types , , and have been obtained, and in the present paper we complete the classification of domestic automorphisms of large spherical buildings of exceptional type of rank at least~ by classifying such automorphisms in the case. Applications of this classification are provided, including Density Theorems showing that each conjugacy class in a group acting strongly transitively on a spherical building intersects a very small number of -cosets, with the stabiliser of a fixed choice of chamber.

Paper Structure

This paper contains 23 sections, 46 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['fixedchambers']}
  3. Geometric preliminaries
  4. Quaternion algebras
  5. Parapolar spaces
  6. Some properties of long root subgroup geometries and metasymplectic spaces
  7. Equator geometries and imaginary lines
  8. Class I automorphisms
  9. Quaternion dual polar spaces of rank 3 in $\mathsf{E_{7,7}}(\mathbb{K})$
  10. The embedding of symplecta in a point residual
  11. The global situation in any point residual
  12. The embedding of $\Gamma$ in $\Delta$
  13. Domesticity of Class I automorphisms
  14. Class II automorphisms
  15. Proof of Theorem \ref{['nofixedchamber']}
  16. ...and 8 more sections

Key Result

Theorem A

Let $\theta$ be an automorphism of the building $\mathsf{E}_8(\mathbb{K})$ with $|\mathbb{K}|> 2$, and suppose that $\theta$ fixes no chamber. The following are equivalent. Also, the building $\mathsf{E_8}(\mathbb{K})$ admits an automorphism of Class I if, and only if, $\mathbb{K}$ admits an associative quadratic division algebra of dimension $4$, whereas it admits an automorphism of Class II if,

Figures (2)

  • Figure 1: The opposition diagrams of type $\mathsf{E_8}$
  • Figure 2: The Dynkin diagram of type $\mathsf{F_4}$ with Bourbaki labelling

Theorems & Definitions (88)

  • Theorem A
  • Theorem B
  • Corollary C
  • Remark 1.1
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['fixedchambers']}
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 78 more