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

TL;DR

This paper computes the representation zeta functions of specific finitely generated nilpotent groups linked to unipotent group schemes over number field rings, using p-adic integrals related to prehomogeneous vector spaces.

Contribution

It introduces a novel method for calculating zeta functions of nilpotent groups via p-adic integrals tied to prehomogeneous vector spaces.

Findings

Explicit formulas for representation zeta functions of studied groups

Connection established between zeta functions and rank varieties of matrices

Method demonstrated for groups associated with certain Lie lattices

Abstract

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.