Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces

We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating $\mathfrak{p}$-adic integrals associated to certain rank varieties of matrices of linear forms.