Arc coordinates for maximal representations

Abstract

We generalize arc coordinates for maximal representations from a hyperbolic surface with boundary into $\text{PSp}(4,\mathbb{R})$, focusing on the case where the surface is a pair of pants. We introduce geometric parameters within the space of right-angled hexagons in the Siegel space $\mathcal{X}$. These parameters enable the visualization of a right-angled hexagon as a polygonal chain inside the hyperbolic plane $\mathbb{H}^{2}$. We explore the geometric properties of reflections in $\mathcal{X}$ and introduce the notion of maximal representation of the reflection group $W_{3}=\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}$. We parametrize maximal representations from $W_{3}$ into $\text{PSp}^{\pm}(4,\mathbb{R})$, this induces a natural parametrization of a subset of maximal and Shilov hyperbolic representations into $\text{PSp}(4,\mathbb{R})$.