Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representation zeta functions of split extensions of $SL_2^m(O)$

J. Moritz Petschick, Margherita Piccolo

TL;DR

The paper studies representation growth for split extensions G = H ⋉_σ O^n of SL_2^m(O) by developing a product formula ζ_G(s) = ζ_H(s) · ζ^G_H(s−1) and applying p-adic integration alongside Mackey theory to obtain explicit zeta functions for new families of groups. It identifies thetyspectral subgroups and provides criteria and proofs showing that, in general, subgroups may differ from the whole group by a constant factor, yet not all subgroups share this property. The authors compute precise zeta functions for two large families of semidirect products, revealing unbounded polynomial representation growth as the representation degree parameter n grows, with explicit formulas for both diagonal natural-module actions and Sym^2 actions. They also provide a non-thetyspectral example, illustrating the limitations of the spectral-constant phenomenon and highlighting the nuanced structure of representation growth in high-dimensional p-adic analytic groups.

Abstract

We consider the representation growth of split extensions of $SL_2^m(O)$. We prove that the corresponding representation zeta functions factor as a product of the representation zeta function of $SL_2^m(O)$ and the relative representation zeta function associated to the extension. We make use of our result by computing the zeta functions for two infinite families of split extensions of $SL_2^m(O)$ explicitly. Along the way, we compute the representation zeta functions of a large class of subgroups of $SL_2^m(O)$.

Representation zeta functions of split extensions of $SL_2^m(O)$

TL;DR

The paper studies representation growth for split extensions G = H ⋉_σ O^n of SL_2^m(O) by developing a product formula ζ_G(s) = ζ_H(s) · ζ^G_H(s−1) and applying p-adic integration alongside Mackey theory to obtain explicit zeta functions for new families of groups. It identifies thetyspectral subgroups and provides criteria and proofs showing that, in general, subgroups may differ from the whole group by a constant factor, yet not all subgroups share this property. The authors compute precise zeta functions for two large families of semidirect products, revealing unbounded polynomial representation growth as the representation degree parameter n grows, with explicit formulas for both diagonal natural-module actions and Sym^2 actions. They also provide a non-thetyspectral example, illustrating the limitations of the spectral-constant phenomenon and highlighting the nuanced structure of representation growth in high-dimensional p-adic analytic groups.

Abstract

We consider the representation growth of split extensions of . We prove that the corresponding representation zeta functions factor as a product of the representation zeta function of and the relative representation zeta function associated to the extension. We make use of our result by computing the zeta functions for two infinite families of split extensions of explicitly. Along the way, we compute the representation zeta functions of a large class of subgroups of .
Paper Structure (13 sections, 15 theorems, 99 equations, 1 figure)

This paper contains 13 sections, 15 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Representation theory
  5. Representation zeta functions
  6. Saturable and uniformly potent pro-p groups
  7. The p-adic integration formalism
  8. Zeta functions of subgroups and semi-direct products
  9. Thetyspectral groups
  10. Lemmata on potency
  11. Product formula for representation zeta functions of semi-direct products
  12. Uniformly potent pro-p groups of large representation growth
  13. A non-thetyspectral subgroup

Key Result

Theorem 1.1

Let $\mathop{\mathrm{\mathcal{O}}}\nolimits$ be a compact discrete valuation ring of characteristic $0$ and residue characteristic $p$. Let $\mathfrak{k}$ be a $3$-dimensional simple $\mathop{\mathrm{\mathcal{O}}}\nolimits$-Lie lattice and let $\mathfrak{h}$ be an open $\mathop{\mathrm{\mathcal{O}}} where $\zeta^{G}_{H}(s)$ is the zeta function associated to the representation $\mathop{\mathrm{Ind

Figures (1)

  • Figure 1: A sketch of the partition of $K^2$ used in the proof of \ref{['prop:non-thety']}, depicting the case $k = 4$. Every circle represents a set of fixed valuation vector, i.e. of the form $p^a\mathop{\mathrm{\mathcal{O}}}\nolimits^\times \times p^b\mathop{\mathrm{\mathcal{O}}}\nolimits^\times$ of $K^2$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 18 more