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

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 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.