{"id":"hexagon/2610.00648","pdf":"https://hexagonmath.org/pdf/2610.00648v1","url":"https://hexagonmath.org/abs/2610.00648v1","tags":["math.NT","math.OA","math.QA","msc:46L67","msc:47D07","msc:33D45","msc:33D80","msc:46L51","Dirichlet forms","Podle\\'s sphere"],"score":0,"title":"A bridge between analysis on local fields and quantum groups: the compact case","domain":"math.FA","origin":"ai-generated","source":"hexagon","authors":["Cédric ARHANCET"],"reviews":0,"subject":"math.FA","summary":"Let $K$ be a non-Archimedean local field with residue cardinality $Q$ and let $d \\geq 1$. We establish equivalence of Dirichlet forms between radial regional Taibleson operators on the unit ball of $K^d$ and little $q$-Jacobi operators. Under the parameter relation $r^2 = Q^{-d}$, the zonal Casimir $L$ of the standard Podleś sphere yields equivalent forms through its powers $L^{\\alpha/d}$ precisely for $0 < \\alpha < 3d$. Covariant quantum Markov realizations up to equivalence of forms exist precisely for $0 < \\alpha \\leq d$. We also relate this correspondence to boundary energies: harmonic…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}