Representation zeta function of a family of maximal class groups: non-exceptional primes (Q6611929)

This article is concerned with representation zeta functions and their \(p\)-local analogues for a certain family \(M_n\) of finitely generated torsion-free nilpotent groups of maximal class (equal to \(n \in \mathbb{N}\)). For a finitely generated torsion-free nilpotent group and a positive integer \(d\), there are finitely many equivalence classes of irreducible complex representations of dimension \(d\) with respect to isomorphism and twisting by a one-dimensional representation by a result of [\textit{A. Lubotzky} and \textit{A. R. Magid}, Varieties of representations of finitely generated groups. Providence, RI: American Mathematical Society (AMS) (1985; Zbl 0598.14042)], and the representation zeta function assembles the numbers of these equivalence classes into a Dirichlet generating function as in [\textit{A. Stasinski} and \textit{C. Voll}, Am. J. Math. 136, No. 2, 501--550 (2014; Zbl 1286.11140)]. Here, the author computes \(p\)-local analogues of the representation zeta function of \(M_n\) for all primes \(p>n\) by explicitly constructing a representative for every equivalence class of irreducible complex representations whose dimension is a power of \(p\). He also obtains the global representation zeta functions for the groups \(M_2\), \(M_3\) and \(M_4\).