Accidental CR structures

TL;DR

This paper investigates the realization of exceptional Lie groups as CR automorphism groups of certain geometric structures, clarifying dimension discrepancies and providing explicit examples of higher codimension CR structures with exceptional symmetries.

Contribution

It constructs explicit CR structures of higher codimension realizing real forms of E6 with exceptional symmetries, resolving previous dimension discrepancies and classifying such structures.

Findings

Realization of E_{II} and E_{III} in dimension 24 as CR automorphism groups.

Complete classification of CR structures with exceptional symmetries and their embeddings.

Identification of 'accidental' CR structures with simple automorphism groups.

Abstract

We noticed a discrepancy between \'Elie Cartan and Sigurdur Helgason about the lowest possible dimension in which the simple exceptional Lie group ${\bf E}_8$ can be realized. This raised the question about the lowest dimensions in which various real forms of the exceptional groups ${\bf E}_\ell$ can be realized. Cartan claims that ${\bf E}_6$ can be realized in dimension 16. However Cartan refers to the complex group ${\bf E}_6$, or its split real form $E_I$. His claim is also valid in the case of the real form denoted by $E_{IV}$. We find however that the real forms $E_{II}$ and $E_{III}$ of ${\bf E}_6$ can not be realized in dimension 16 \`a la Cartan. In this paper we realize them in dimension 24 as groups of CR automorphisms of certain CR structures of higher codimension. As a byproduct of these two realizations, we provide a full list of CR structures $(M,H,J)$ and their CR embeddings in an appropriate ${\bf C}^N$, which satisfy the following conditions: (1) they have real codimension $k>1$, (2) the real vector distribution $H$ proper for the action of the complex structure $J$ is such that $[H,H]+H=TM$, (3) the local group $G_J$ of CR automorphisms of the structure $(M,H,J)$ is simple, acts transitively on $M$ and has isotropy $P$ being a parabolic subgroup in $G_J$, (4) the local symmetry group $G$ of the vector distribution $H$ on $M$ coincides with the group $G_J$ of CR automorphisms of $(M,H,J)$. Because all the CR structures from our list satisfy the last property we call them accidental. Our CR structures of higher codimension with the exceptional symmetries $E_{II}$ and $E_{III}$ are particular entries in this list.