Arc coordinates for maximal representations

We generalize arc coordinates for maximal representations from a hyperbolic surface with boundary into PSp(4,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 X. These parameters enable the visualization of a right-angled hexagon as a polygonal chain inside the hyperbolic plane H². We explore the geometric properties of reflections in X and introduce the notion of maximal representation of the reflection group W₃=Z/2Z*Z/2Z*Z/2Z. We parametrize maximal representations from W₃ into PSp±(4,R), this induces a natural parametrization of a subset of maximal and Shilov hyperbolic representations into PSp(4,R).