Zeta functions associated to admissible representations of compact p-adic Lie groups

Let $G$ be a profinite group. A strongly admissible smooth representation $\rho$ of $G$ over $\mathbb{C}$ decomposes as a direct sum $\rho \cong \bigoplus_{\pi \in \mathrm{Irr}(G)} m_\pi(\rho) \, \pi$ of irreducible representations with finite multiplicities $m_\pi(\rho)$ such that for every positive integer $n$ the number $r_n(\rho)$ of irreducible constituents of dimension $n$ is finite. Examples arise naturally in the representation theory of reductive groups over non-archimedean local fields. In this article we initiate an investigation of the Dirichlet generating function $$\zeta_\rho (s) = \sum_{n=1}^\infty r_n(\rho) n^{-s} = \sum_{\pi \in \mathrm{Irr}(G)} \frac{m_\pi(\rho)}{(\dim \pi)^s}$$ associated to such a representation $\rho$. Our primary focus is on representations $\rho = \mathrm{Ind}_H^G(\sigma)$ of compact $p$-adic Lie groups $G$ that arise from finite dimensional representations $\sigma$ of closed subgroups $H$ via the induction functor. In addition to a series of foundational results - including a description in terms of $p$-adic integrals - we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro-$p$ groups. A key ingredient of our proof is Hironaka's resolution of singularities, which yields formulae of Denef-type for the relevant zeta functions. In some detail, we consider representations of open compact subgroups of reductive $p$-adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees and (ii) the $p$-adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.