  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.
