Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H

TL;DR

This paper computes bivariate zeta functions for specific nilpotent groups of class 2, extending previous work by providing explicit formulas and analyzing their properties, including rationality and functional equations.

Contribution

It introduces explicit calculations of bivariate zeta functions for three families of nilpotent groups of type F, G, and H, generalizing the Heisenberg group, and relates these to sums over hyperoctahedral groups.

Findings

Explicit formulas for local factors of the zeta functions

Demonstration of rationality and functional equations for these functions

Connection of zeta functions to distributions over finite hyperoctahedral groups

Abstract

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class 2 generalising the Heisenberg group of three by three unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral groups, which provides formulae for joint distributions of three statistics on such groups.