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hexagon:2610.00649v1

math.FA — Functional Analysis

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

Contributed by Cédric ARHANCET

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.

A mix of human-written and AI-generated textHuman understanding: some parts$q$-Hankel transformDirichlet formsHahn--Exton $q$-Bessel functionsPoisson kernelsTaibleson operatorlocal fieldsquantum Markov semigroupsquantum group $E(2)$
Oct 29, 2026PDF ↗View work →
hexagon:2610.00648v1

math.FA — Functional Analysis

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

Contributed by Cédric ARHANCET

Let KK be a non-Archimedean local field with residue cardinality QQ and let d≥1d \geq 1. We establish equivalence of Dirichlet forms between radial regional Taibleson operators on the unit ball of KdK^d and little qq-Jacobi operators. Under the parameter relation r2=Q−dr^2 = Q^{-d}, the zonal Casimir LL of the standard Podleś sphere yields equivalent forms through its powers Lα/dL^{\alpha/d} precisely for 0<α<3d0 < \alpha < 3d. Covariant quantum Markov realizations up to equivalence of forms exist precisely for 0<α≤d0 < \alpha \leq d. We also relate this correspondence to boundary energies: harmonic extensions on the tree of balls recover a Taibleson energy, while the energy per level of quantum Poisson integrals converges to a multiple of the Casimir energy. Finally, we extend the correspondence and the sharp covariance threshold to quantum projective spaces, identifying the Taibleson orders associated with their zonal Casimir operators.

A mix of human-written and AI-generated textHuman understanding: some partsDirichlet formsPodle\'s spherePoisson boundariesTaibleson operatorlocal fieldsnoncommutative $\L^p$-spacesquantum Markov semigroupsquantum groupsquantum projective spacesrandom walks on trees
Oct 29, 2026PDF ↗View work →

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