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- W2606494943 abstract "We compute explicit formulae for Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of potent and saturable principal congruence subgroups of $mathrm{SL}_4^m(mathfrak{o})$ ($minmathbb{N}$) for $mathfrak{o}$ a compact DVR of characteristic $0$ and odd residue field characteristic. In doing so we develop a novel method for computing Poincare 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 $4times 4$ 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." @default.
- W2606494943 created "2017-04-28" @default.
- W2606494943 creator A5035893794 @default.
- W2606494943 date "2017-04-13" @default.
- W2606494943 modified "2023-09-27" @default.
- W2606494943 title "Poincar'e series of Lie lattices and representation zeta functions of arithmetic groups" @default.
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