Poincar\'e series of Lie lattices and representation zeta functions of arithmetic groups

Abstract

We compute explicit formulae for Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of potent and saturable principal congruence subgroups of $\mathrm{SL}_4^m(\mathfrak{o})$ ($m\in\mathbb{N}$) for $\mathfrak{o}$ a compact DVR of characteristic $0$ and odd residue field characteristic. In doing so we develop a novel method for computing Poincar\'e series associated with commutator matrices of $\mathfrak{o}$-Lie lattices with finite abelianization and whose rank-loci enjoy an additional smoothness property. We give explicit formulae for the abscissa of convergence of the representation zeta functions of potent and saturable FAb $p$-adic analytic groups whose associated Lie lattices satisfy the hypotheses of the aforementioned method. As a by-product of our computations we find that not all $4\times 4$ traceless matrices over a finite quotient of $\mathfrak{o}$ admit shadow-preserving lifts, thus disproving that smooth loci of constant centralizer dimension in $\mathfrak{sl}_4(\mathbb{C})$ ensure presence of shadow-preserving lifts for almost all primes as suggested in a previous paper by Avni, Klopsch, Onn and Voll.