Volume invariant and maximal representations of discrete subgroups of Lie groups

Let $\mathrm{\Gamma }$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\mathrm{\Gamma }$ into $G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\mathrm{\Gamma }$ into $G$ except for $\Gamma\subset $\mathrm{P}\mathrm{S}{\mathrm{L}}_2\mathbb{C}$$ a nonuniform lattice.