Let @math be a number field with ring of integers @math . We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups @math that are indexed by rational primes which are unramified in @math . We show that these local zeta functions satisfy functional equations upon the inversion of the prime.
In the recent past, zeta functions associated to Heisenberg groups and their various generalizations have often served as a test case for an ensuing general theory. For instance, the seminal paper @cite_0 contains special cases of the computations done in the present paper as examples. Similarly, Ezzat @cite_9 computed the representation zeta functions of the groups @math for quadratic number rings @math , enumerating irreducible finite-dimensional complex representations of such groups up to twists by one-dimensional representations. The paper @cite_2 develops a general framework for the study of representation zeta functions of finitely generated nilpotent groups. Moreover, it generalized Ezzat's explicit formulae to arbitrary number rings and more general group schemes.
@cite_0 @cite_9 @cite_2
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We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and @math -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- @math -nilpotent groups obtained from three infinite families of group schemes, generalizing 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 @math . 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 @math -adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces. Abstract We calculate the representation zeta function of the Heisenberg group over the integers of a quadratic number field. In general, the representation zeta function of a finitely generated torsion-free nilpotent group enumerates equivalence classes of representations, called twist-isoclasses. This calculation is based on an explicit description of a representative from each twist-isoclass. Our method of construction involves studying the eigenspace structure of the elements of the image of the representation and then picking a suitable basis for the underlying vector space. We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively. 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. These zeta functions encode, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. In this paper, we show that such bivariate zeta functions specialise to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialise to (univariate) twist representation zeta functions. Moreover, we show that these zeta functions satisfy Euler factorisations and almost all of their Euler factors satisfy rationality and functional equations. In the second part of this work, we compute the above mentioned zeta functions of three infinite families of nilpotent groups of class two and deduce some combinatorial corollaries.