Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B

TL;DR

This paper investigates the representation zeta functions of certain nilpotent groups, proving their rationality, functional equations, and explicit formulas, while connecting these to Weyl group statistics and p-adic integrals.

Contribution

It provides explicit formulas for zeta functions of nilpotent groups, linking them to Weyl group generating functions and p-adic integrals, extending previous results.

Findings

Proved rationality and functional equations for local zeta factors.

Derived explicit formulas involving Dedekind zeta functions.

Connected zeta functions to Weyl group statistics and p-adic integrals.

Abstract

We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and p-adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class-2-nilpotent groups obtained from three infinite families of group schemes, generalising the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type B. This allows us to establish a formula for the joint distribution of three functions, or 'statistics', on such Weyl groups. Finally, we compare our explicit formulae to p-adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces.