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Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type

Sira Busch, Jeroen Schillewaert, Hendrik Van Maldeghem

TL;DR

The article studies how Levi subgroups of parabolic subgroups act on residues of irreducible spherical buildings of simply laced type, by introducing special and general projectivity groups for simplices. It provides two parallel approaches: an algebraic route via Levi decompositions in Chevalley groups and a geometric route via root elations and projections, culminating in a complete determination of when the two projectivity groups coincide and how Levi actions are realized on residues. The main contributions include MR0–MR4, which relate projectivity groups to stabilizers on residues, classify exceptional cases for $ extsf{E}_6, extsf{E}_7, extsf{E}_8, extsf{D}_n$, and establish a connectivity tool for opposite chambers; the results yield precise geometric incarnations of Levi subgroups and reveal when polar types govern the coincidence of special and general projectivity groups. The findings advance the understanding of how incidence-geometry and algebraic-group structures intertwine in spherical buildings, with implications for non-simply laced and non-split settings and potential applications to the theory of Lie-type groups over arbitrary fields. Overall, the paper provides a comprehensive, tabled map of projectivity groups across irreducible residues, illuminating the Levi-action on residues and enriching the geometric toolkit for studying groups of Lie type via buildings.

Abstract

We introduce the special and general projectivity groups attached to a simplex $F$ of a thick irreducible spherical building of simply laced type. If the residue of $F$ is irreducible, we determine the permutation group of both projectivity groups of $F$, acting on the residue of $F$ and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types $\mathsf{E_6,E_7,E_8}$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide.

Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type

TL;DR

The article studies how Levi subgroups of parabolic subgroups act on residues of irreducible spherical buildings of simply laced type, by introducing special and general projectivity groups for simplices. It provides two parallel approaches: an algebraic route via Levi decompositions in Chevalley groups and a geometric route via root elations and projections, culminating in a complete determination of when the two projectivity groups coincide and how Levi actions are realized on residues. The main contributions include MR0–MR4, which relate projectivity groups to stabilizers on residues, classify exceptional cases for , and establish a connectivity tool for opposite chambers; the results yield precise geometric incarnations of Levi subgroups and reveal when polar types govern the coincidence of special and general projectivity groups. The findings advance the understanding of how incidence-geometry and algebraic-group structures intertwine in spherical buildings, with implications for non-simply laced and non-split settings and potential applications to the theory of Lie-type groups over arbitrary fields. Overall, the paper provides a comprehensive, tabled map of projectivity groups across irreducible residues, illuminating the Levi-action on residues and enriching the geometric toolkit for studying groups of Lie type via buildings.

Abstract

We introduce the special and general projectivity groups attached to a simplex of a thick irreducible spherical building of simply laced type. If the residue of is irreducible, we determine the permutation group of both projectivity groups of , acting on the residue of and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types . Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of coincide.
Paper Structure (26 sections, 38 theorems, 38 equations, 2 tables)

This paper contains 26 sections, 38 theorems, 38 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of the Main Results
  3. Spherical buildings
  4. Groups of projectivity
  5. The Levi decomposition in Chevalley groups
  6. Main results
  7. Lie incidence geometries
  8. Point-line geometries, projective spaces, polar spaces and parapolar spaces
  9. A connectivity theorem
  10. General observations and proof of \ref{['MR0']}
  11. Projective spaces
  12. Proof of \ref{['MR1']}
  13. Proof of \ref{['MR2']}
  14. Type $\mathsf{E_6}$
  15. Type $\mathsf{E_7}$
  16. ...and 11 more sections

Key Result

Proposition 2.1

Let $F$ and $F'$ be opposite simplices of a spherical building $\Delta$. Let $v$ be a vertex of $\Delta$ such that $v\sim F$, and set $i:=\mathop{\mathrm{\mathsf{typ}}}\nolimits(v)\in I$. Then the type $i'$ of the vertex $\mathsf{proj}^{F}_{F'}(v)$ is the opposite in $\mathsf{Res}(F')$ of the opposi

Theorems & Definitions (66)

  • Proposition 2.1: Proposition 3.29 of Tits:74
  • Lemma 2.2
  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary 1
  • Theorem D
  • Theorem E
  • Lemma 2.3
  • proof
  • ...and 56 more