Univariate and bivariate zeta functions of unipotent group schemes of type $G$

TL;DR

This paper computes zeta functions related to certain nilpotent groups of class 2, generalizing previous work on unipotent group schemes, and applies these results to distributions on Weyl groups of type B.

Contribution

It introduces a generalization of unipotent group schemes and derives explicit formulas for their representation and class counting zeta functions.

Findings

Explicit formulas for univariate and bivariate zeta functions of the groups.

Connection between zeta functions and distributions on Weyl groups of type B.

Extension of previous work by Stasinski, Voll, and Lins.

Abstract

We compute the representation and class counting zeta functions for a family of torsion-free finitely generated nilpotent groups of nilpotency class $2$. These groups arise from a generalisation of one the families of unipotent groups schemes treated by Stasinski and Voll, and Lins. The univariate zeta functions are obtained by specialising the respective bivariate zeta functions defined by Lins. These are also used to deduce a formula for a joint distribution on Weyl groups of type $B$.