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

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