Volume invariant and maximal representations of discrete subgroups of Lie groups

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

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