Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes
The main topic of this doctoral thesis is zeta functions of groups. Let G be a unipotent group scheme defined over the ring of integers O of a number field. The group G(O) of O-rational points is a finitely generated torsion-free nilpotent group. We introduce two bivariate zeta functions related to groups of the form G(O): firstly the bivariate representation zeta function of G(O), which enumerates the isomorphism classes of irreducible complex representations of finite dimensions of its
... sions of its congruence quotients, and secondly the bivariate conjugacy class zeta function of G(O), which enumerates the conjugacy classes of each size of its congruence quotients. These zeta functions might be used as tools for understanding another (univariate) zeta functions, as they both specialise to class number zeta functions, which enumerate class numbers of the congruence quotients. Additionally, in case of nilpotency class two, bivariate representation zeta functions specialise to twist representation zeta functions, which are zeta functions enumerating the irreducible complex characters of finite dimensions up to tensoring by one-dimensional characters. We show that bivariate representation and bivariate conjugacy class zeta functions satisfy Euler decompositions and that almost all of their Euler factors are rational and satisfy functional equations. We also prove that they converge on some domains and, furthermore, their maximal domains of convergence and meromorphic continuation are independent of the number field O considered, up to finitely many local factors. We provide formulae for the bivariate zeta functions of three infinite families of groups of nilpotency class two of the form G(O) which generalise the Heisenberg group of 3 x 3-unitriangular matrices over O. As an application, we establish formulae for the joint distributions of three statistics on finite hyperoctahedral groups.