The gap phenomenon in parabolic geometries

TL;DR

This paper investigates the maximum possible symmetry dimensions in non-flat parabolic geometries, establishing universal bounds and demonstrating their sharpness across various Lie group cases.

Contribution

It derives a universal upper bound on submaximal symmetry dimensions for parabolic geometries using Tanaka theory and confirms sharpness through explicit models.

Findings

Universal upper bound on submaximal symmetry dimension derived.

Bound is sharp in almost all complex and split-real cases.

Explicit computations for all simple Lie groups provided.

Abstract

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant's version of the Bott-Borel-Weil theorem to show that this bound is in fact sharp in almost all complex and split-real cases by exhibiting (abstract) models. We explicitly compute all submaximal symmetry dimensions when G is any complex or split-real simple Lie group.