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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Zeta functions associated to admissible representations of compact $p$-adic Lie groups
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by Steffen Kionke and Benjamin Klopsch PDF
Trans. Amer. Math. Soc. 372 (2019), 7677-7733 Request permission

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.

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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
  • Email: steffen.kionke@kit.edu
  • Benjamin Klopsch
  • Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225 Düsseldorf, Germany
  • MR Author ID: 658412
  • Email: klopsch@math.uni-duesseldorf.de
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7677-7733
  • MSC (2010): Primary 20E18; Secondary 20C15, 20G25, 22E50, 11M41
  • DOI: https://doi.org/10.1090/tran/7834
  • MathSciNet review: 4029678