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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Steffen Kionke and Benjamin Klopsch
Journal: Trans. Amer. Math. Soc. 372 (2019), 7677-7733
MSC (2010): Primary 20E18; Secondary 20C15, 20G25, 22E50, 11M41
Published electronically: April 3, 2019
MathSciNet review: 4029678
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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.

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Additional Information

Steffen Kionke
Affiliation: Karlsruhe Institute of Technology, Institute for Algebra and Geometry, Englerstrasse 2, 76131 Karlsruhe, Germany
MR Author ID: 1070549

Benjamin Klopsch
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225 Düsseldorf, Germany
MR Author ID: 658412

Received by editor(s): August 22, 2017
Received by editor(s) in revised form: August 5, 2018, and January 14, 2019
Published electronically: April 3, 2019
Additional Notes: We acknowledge support by DFG grant KL 2162/1-1.
Article copyright: © Copyright 2019 American Mathematical Society