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 $W_3=\mathbb{Z}/2\mathbb{Z}\ast \mathbb{Z}/2\mathbb{Z}\ast \mathbb{Z}/2\mathbb{Z}$. 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 $\text{PSp}(4,\mathbb{R})$.