Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups

Abstract

Abstract. Let G be a finitely generated nilpotent group. The representation zeta function ζ G (s) of G enumerates twist isoclasses of finite-dimensional irreducible complex representations of G. We prove that ζ G (s) has rational abscissa of convergence α(G) and may be meromorphically continued to the left of α(G) and that, on the line {s ∈ C | Re(s) = α(G)}, the continued function is holomorphic except for a pole at s = α(G). A Tauberian theorem yields a precise asymptotic result on the representation growth of G in terms of the position and order of this pole.We obtain these results as a consequence of a result establishing uniform analytic properties of representation zeta functions of torsion-free finitely generated nilpotent groups of the form G(O), where G is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring O of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of G, independent of O.

Zeta functions associated to admissible representations of compact 𝑝-adic Lie groups

“…Dirichlet generating functions have also been employed to study the distribution of finite-dimensional irreducible representations of finitely generated nilpotent groups. For such a group Γ one defines and studies the zeta function enumerating twist-isoclasses of irreducible representations of Γ; for instance, see [33,56,49,21]. Many theorems on representation zeta functions, e.g., regarding rationality, pole spectra, and functional equations, have analogues in the twist-isoclass setting; the Kirillov orbit method can be adjusted to take into account twist-isoclasses and thus yields a basic tool.…”

“…In recent years the subject of representation growth has advanced with a primary focus on zeta functions enumerating (i) irreducible representations of arithmetic lattices and compact p-adic Lie groups, (ii) twist-isoclasses of irreducible representations of finitely generated nilpotent groups; for instance, see [41,3,4,5,1] and [58,56,49,21,33,50], or the relevant surveys [38,59]. The aim of this paper is to introduce and study a new, more general zeta function that can be associated to any 'suitably tame' infinite-dimensional representation of a group.…”

The average size of the kernel of a matrix and orbits of linear groups

“…In the present context, this process can be formalised just as in the case of representation zeta functions of unipotent groups (see [79, § 7]). Moreover, Proposition 4.17 and a variation of [79, proof of Theorem 7.3] (which relies heavily on arguments due to Duong and Voll [33]) show that in fact Z red M (T ) = 1/(1 − T ) for any M . This is intuitively plausible: ifM ⊂ M d×e (O n ) is a submodule, then the group (O/P) × acts freely onM \ {0} and preserves kernels whence |M | · ask(M ) ≡ |Ṽ | (mod (q − 1)), whereṼ = O d n .…”

Computing local zeta functions of groups, algebras, and modules

“…Proof. In the setting of [19,Prop. 3.4], the rational numbers A j and B j can actually be assumed to be integers; this follows e.g.…”