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A bridge between analysis on local fields and quantum groups: the noncompact case

A mix of human-written and AI-generated textHuman understanding: some partsmath.FA — Functional Analysismath.NT — Number Theorymath.OA — Operator Algebrasmath.QA — Quantum Algebra

Contributed by Cédric ARHANCET ↗

Version 1 / Oct 29, 2026 / CC BY-SA 4.0

Abstract

Let KK be a non-Archimedean local field with residue cardinality QQ and let d≥1d \geq 1. We establish a radial correspondence between analysis on KdK^d and Woronowicz's quantum group Er(2)\mathrm{E}_r(2), with r2=Q−dr^2 = Q^{-d}. For every α>0\alpha > 0, the radial Taibleson operator of order α\alpha is unitarily equivalent, through a unitary commuting with dilations, to a bilateral birth–death operator with Hahn–Exton qq-Bessel eigenfunctions. Their Dirichlet forms are equivalent on the same weighted sequence space, with optimal comparison constants. For α=d\alpha = d, the bilateral operator is the radial part of a self-adjoint Casimir on the quantum plane Er(2)/T\mathrm{E}_r(2)/\mathbb{T}, whose heat semigroup is covariant and completely positive. We further obtain these quantum-plane dynamics by contraction of the compact Casimir dynamics on the Podleś sphere, with Mosco convergence of the forms and pointwise norm convergence of the compressed heat maps.

Provenance statement

The main mathematical ideas and the overall research direction originated with the author. AI tools contributed to the development of some calculations and arguments. The author considers the manuscript broadly correct, but further detailed checking and several rounds of revision and polishing are still needed. This version is therefore shared as a work in progress, rather than as a fully checked final manuscript. The author takes responsibility for its content and welcomes corrections and comments.

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Version history

  1. v1Submitted by Cédric ARHANCETInitial depositCurrentOct 29, 2026