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

Abstract:

Let $G$ be a finitely generated nilpotent group. The representation zeta function $\zeta _G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\zeta _G(s)$ has rational abscissa of convergence $\alpha (G)$ and may be meromorphically continued to the left of $\alpha (G)$ and that, on the line $\{s\in \mathbb {C} \mid \mathrm {Re}(s) = \alpha (G)\}$, the continued function is holomorphic except for a pole at $s=\alpha (G)$. 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 $\mathbf {G}(\mathcal {O})$, where $\mathbf {G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal {O}$ 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 $\mathbf {G}$, independent of $\mathcal {O}$.