The orbital counting problem for hyperconvex representations

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 $\textrm{PSL}(2,\mathbb R)\times\textrm{PSL}(2,\mathbb R),$ induced by choosing two points on the Teichmüller space of the surface; and representations on the Hitchin component of $\textrm{PSL}(d,\mathbb R).$ We also prove a mixing property for the Weyl chamber flow in this setting.

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