Configurations of flags in orbits of real forms

TL;DR

This paper investigates flag configurations in closed orbits of real forms using GIT tools and applies cross ratio coordinates to identify boundary unipotent representations of the figure eight knot complement's fundamental group into real forms of PGL(4,C).

Contribution

It introduces a GIT-based approach to study flag configurations in real form orbits and applies cross ratio coordinates to classify boundary unipotent representations.

Findings

Identified boundary unipotent representations into real forms of PGL(4,C).

Developed a GIT framework for flag configurations in real form orbits.

Used cross ratio coordinates for generic configurations.

Abstract

In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of $\mathrm{PGL}(4,\mathbb{C})$.