Lattices in \(\mathbb {R}^n\rtimes \textrm{SL}_2(\mathbb {R})\)

We determine the existence of cocompact lattices in groups of the form \(\textrm{V}\rtimes \textrm{SL}_2(\mathbb {R})\) $\text{V}⋊{\text{SL}}_2(\mathbb{R})$ , where \(\textrm{V}\) $\text{V}$ is a finite dimensional real representation of \(\textrm{SL}_2(\mathbb {R})\) ${\text{SL}}_2(\mathbb{R})$ . It turns out that the answer depends on the parity of \(\dim (\textrm{V})\) $dim(\text{V})$ when the representation is irreducible.