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

Let $K$ be a non-Archimedean local field with residue cardinality $Q$ and let $d \geq 1$. We establish a radial correspondence between analysis on $K^d$ and Woronowicz's quantum group $\mathrm{E}_r(2)$, with $r^2 = Q^{-d}$. For every $\alpha > 0$, the radial Taibleson operator of order $\alpha$ is unitarily equivalent, through a unitary commuting with dilations, to a bilateral birth–death operator with Hahn–Exton $q$-Bessel eigenfunctions. Their Dirichlet forms are equivalent on the same weighted sequence space, with optimal comparison constants. For $\alpha = d$, the bilateral operator is the radial part of a self-adjoint Casimir on the quantum plane $\mathrm{E}_r(2)/\mathbb{T}$, 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.