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Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units

Reinier Sorgdrager

TL;DR

The paper proves the sharp bound $ ext{GKdim}( ext{Pi})\le [K:\mathbf Q_p]$ for admissible $L$-Banach representations of $\text{GL}_2K$ or quaternion units $D^ imes$ when the locally analytic vectors admit an infinitesimal character, for $p>2$. The approach blends filtrations on distribution algebras, Lazard’s isomorphism, and a Casimir-ideal annihilation mechanism acting on associated graded modules to control the Krull dimension of the reduction $\mathop{\mathrm{gr}}\overline M$, tying representation-theoretic invariants to Lie-theoretic central elements. The result sharpens previous bounds in the $p$-adic Langlands context and yields optimal bounds in the $ ext{GL}_2$ case, with indications that the method could generalize to higher rank groups. The work also provides a conceptual explanation for the Casimir-like ideals that govern the upper bounds and discusses potential extensions to families and ramified base fields, as well as connections to patched modules in the $p$-adic Langlands program. Overall, the paper advances a structurally transparent route to GK-bounds via infinitesimal characters and graded-center annihilation, offering tools for future generalizations and applications in $p$-adic representation theory.

Abstract

We prove that an admissible $p$-adic Banach representation of $\text{GL}_2K$ whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension $\leq[K\colon\mathbf Q_p]$, where $p>2$ and $K$ is a $p$-adic field. We also prove this for the group of units of the quaternions over $K$ replacing $\text{GL}_2K$. In the process, we make some observations in the theory of $p$-adic Banach representations that might be of independent interest.

Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units

TL;DR

The paper proves the sharp bound for admissible -Banach representations of or quaternion units when the locally analytic vectors admit an infinitesimal character, for . The approach blends filtrations on distribution algebras, Lazard’s isomorphism, and a Casimir-ideal annihilation mechanism acting on associated graded modules to control the Krull dimension of the reduction , tying representation-theoretic invariants to Lie-theoretic central elements. The result sharpens previous bounds in the -adic Langlands context and yields optimal bounds in the case, with indications that the method could generalize to higher rank groups. The work also provides a conceptual explanation for the Casimir-like ideals that govern the upper bounds and discusses potential extensions to families and ramified base fields, as well as connections to patched modules in the -adic Langlands program. Overall, the paper advances a structurally transparent route to GK-bounds via infinitesimal characters and graded-center annihilation, offering tools for future generalizations and applications in -adic representation theory.

Abstract

We prove that an admissible -adic Banach representation of whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension , where and is a -adic field. We also prove this for the group of units of the quaternions over replacing . In the process, we make some observations in the theory of -adic Banach representations that might be of independent interest.
Paper Structure (26 sections, 28 theorems, 124 equations)

This paper contains 26 sections, 28 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Acknowledgements
  4. Preliminaries (mostly about filtrations)
  5. Rings and filtrations
  6. Brief reminder on Banach representations
  7. $p$-valued groups and their associated filtered rings
  8. Constructions from finite type $\mathcal{O}_L\llbracket H\rrbracket$-modules
  9. There can be no confusion about $\mathop{\mathrm{gr}}\nolimits\overline M$
  10. Lie theory
  11. Exponential map
  12. Infinitesimal action and character
  13. The $\mathbf Z_p$-Lie algebra $H$
  14. Lazard's isomorphism
  15. Groups of $p$-powers
  16. ...and 11 more sections

Key Result

Theorem 1

Let $p>2$ and let $K$ and $L$ be $p$-adic fields. Let $\Pi$ be an admissible $L$-Banach space representation of either $\mathop{\mathrm{GL}}\nolimits_2K$ or of $D^\times$, where $D$ are the quaternions over $K$. If the locally analytic vectors of $\Pi$ have an infinitesimal character, then $\Pi$ has

Theorems & Definitions (60)

  • Theorem : Theorem \ref{['mainmainthm']}
  • Definition 1.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • ...and 50 more