On uniqueness of submaximally symmetric parabolic geometries

TL;DR

This paper investigates the uniqueness of submaximally symmetric parabolic geometries, establishing local uniqueness results for certain complex or split-real simple Lie groups of rank at least three, and for the specific case of (G_2, P_2).

Contribution

It provides new local uniqueness results for submaximally symmetric structures in parabolic geometries of specific Lie groups, advancing understanding of their symmetry gap.

Findings

Established local uniqueness for submaximal symmetry in complex or split-real simple Lie groups of rank ≥ 3.

Proved local uniqueness for the (G_2, P_2) parabolic geometry.

Clarified the structure of symmetry gaps in these geometries.

Abstract

Among (regular, normal) parabolic geometries of type $(G,P)$, there is a locally unique maximally symmetric structure and it has symmetry dimension $\dim(G)$. The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When $G$ is a complex or split-real simple Lie group of rank at least three or when $(G,P) = (G_2,P_2)$, we establish a local uniqueness result for submaximally symmetric structures of type $(G,P)$.