Poincaré series of Lie lattices and representation zeta functions of arithmetic groups

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 Poincaré 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.