Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties

Abstract

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta functions satisfy Euler factorizations and almost all of their Euler factors are rational and satisfy functional equations. Moreover, we show that such bivariate zeta functions specialize to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialize to (univariate) twist representation zeta functions.

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

“…Moreover, the local factors of both zeta functions satisfy functional equations. This generalises [8,Theorem 1.4] for these groups.…”

“…In the first part [8] of this work, we introduced bivariate zeta functions of groups associated to unipotent group schemes over rings of integers of numbers fields. In this second part, we provide explicit examples of such zeta functions for infinite families of nilpotent groups of class 2, and use these expressions to provide formulae for joint distributions of three statistics on hyperoctahedral groups.…”

The average size of the kernel of a matrix and orbits of linear groups, II: duality

“…Under the aforementioned assumption, both types of conjugacy class zeta functions coincide locally in the sense that ζ cc G⊗ov (s) = ζ cc G(ov) (s) for each v ∈ V k . The conjugacy class zeta functions studied by Berman et al [4], the author [26], and Lins [21,22] satisfy (ii).…”

“…Stasinski and Voll [28, §2.1] studied "unipotent group schemes" (particular types of integral forms of unipotent algebraic groups) defined in terms of nilpotent Lie lattices. Lins [21,22] introduced and studied bivariate conjugacy class and representation zeta functions attached to such group schemes. As explained in [21, §1.2], both of her bivariate zeta functions naturally specialise to the univariate conjugacy class zeta functions from §6.1.…”

“…As explained in [21, §1.2], both of her bivariate zeta functions naturally specialise to the univariate conjugacy class zeta functions from §6.1. For a second proof of Corollary 6.3, we may combine Lins's integral formalism [21, §4.2] and Remark 4.13; see [21,Rem. 4.10].…”