Rationality of twist representation zeta functions of compact \(p\)-adic analytic groups (Q6629469)

Let \(G\) be a group. The zeta series of \(G\) is the formal Dirichlet series \(Z_{G}(s)=\sum_{i=1}^{\infty} r_{n}(G)n^{-s}=\sum_{\rho \in \mathrm{Irr}(G)} \rho(1)^{-s}\) where \(r_{n}(G)\) is the number of isomorphism classes of complex irreducible \(n\)-dimensional representations of \(G\) (assumed to be finite for each \(n\)) and \(\mathrm{Irr}(G)\) is the set of irreducible characters of \(G\). If the sequence \(R_{N}(G) = \sum_{i=1}^{N} r_{i}(G)\) grows at most polynomially, \(Z_{G}(s)\) defines a holomorphic function \(\zeta_{G}(s)\) on some right half-plane of \(\mathbb{C}\), which is called the representation zeta function of \(G\).\N\NThe group \(G\) is called representation rigid if the number \(r_{n}(G)\) is finite, for each \(n\). There are significant classes of groups that are not representation rigid, such as torsion-free nilpotent groups or reductive \(p\)-adic groups with infinitely many 1-dimensional representations, like \(\mathrm{GL}_{n}(\mathbb{Z}_{p})\). These groups therefore do not possess a representation zeta function in the usual sense. Nevertheless, it turns out that in many cases the number \(\widetilde{r}_{n}(G)\) of irreducible representations of dimension \(n\) up to one-dimensional twists (i.e., tensoring by one-dimensional representations) is finite for all \(n\). Such groups are called twist rigid. For a twist rigid group \(G\), one can define the Dirichlet series \(\widetilde{Z}_{G}=\sum_{n=1}^{\infty} \widetilde{r}_{n}(G)n^{-s}\) and its meromorphic continuation \(\widetilde{\zeta}_{G}(s)\) called the twist zeta series and twist zeta function, respectively\N\NIn the paper under review, the author proves Theorem 1.1: Let \(G\) be a twist rigid compact \(p\)-adic analytic group. Then, \(\widetilde{\zeta}_{G}(\)s) is virtually rational in \(p^{-s}\). If in addition \(G\) is pro-\(p\), then \(\widetilde{\zeta}_{G}(s)\) is rational in \(p^{-s}\).\N\NAn interesting consequence of Theorem 1.1 is Corollary 1.2: Let \(G\) be a twist rigid compact \(p\)-adic analytic group. Then the following holds regarding \(\widetilde{\zeta}_{G}(s)\): (i) it extends meromorphically to the entire complex plane; (ii) it has an abscissa of convergence which is a rational number.\N\NTo establish these results, the author develops a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass.