Volume invariant and maximal representations of discrete subgroups of Lie groups

Let $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ into $G$ except for $\Gamma\subset \mathrm{PSL_2 \mathbb{C}}$ a nonuniform lattice.