The orbital counting problem for hyperconvex representations

Abstract

We give a precise counting result on the symmetric space of a noncompact real algebraic semisimple group $G,$ for a class of discrete subgroups of $G$ that contains, for example, representations of a surface group on $\text{PSL}(2,\mathbb{R})\times \text{PSL}(2,\mathbb{R}),$ induced by choosing two points on the Teichmüller space of the surface; and representations on the Hitchin component of $\text{PSL}(d,\mathbb{R}).$ We also prove a mixing property for the Weyl chamber flow in this setting.