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

Abstract:

Let $G$ be a profinite group. A strongly admissible smooth representation $\varrho$ of $G$ over ${\mathbb {C}}$ decomposes as a direct sum $\varrho \cong \bigoplus _{\pi \in \text {Irr}(G)} m_\pi (\varrho ) \pi$ of irreducible representations with finite multiplicities $m_\pi (\varrho )$ such that, for every positive integer $n$, the number $r_n(\varrho )$ of irreducible constituents of dimension $n$ is finite. Examples arise naturally in the representation theory of reductive groups over nonarchimedean local fields. In this article we initiate an investigation of the Dirichlet generating function \[ \zeta _\varrho (s) = \sum \nolimits _{n=1}^\infty r_n(\varrho ) n^{-s} = \sum \nolimits _{\pi \in \text {Irr}(G)} \frac {m_\pi (\varrho )}{(\dim \pi )^s} \] associated to such a representation $\varrho$.

Our primary focus is on representations $\varrho = \text {Ind}_H^G(\sigma )$ 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.