The Haar measure of the p -adic rotation group SO(3) p via nautical angles

Source abstract

We study the explicit construction of the Haar measure on the compact [Formula: see text]-adic rotation group [Formula: see text] by nautical (Cardano) parametrization. Exploiting its topological group isomorphism with [Formula: see text] of [Formula: see text]-adic quaternions modulo scalars, we derive the corresponding change-of-variable formulas and compute the associated Jacobian in the [Formula: see text]-adic setting, which we combine with the known Haar measure on the multiplicative group of [Formula: see text]-adic quaternions [Formula: see text]. This yields an explicit formula for the normalized Haar measure on [Formula: see text] in nautical coordinates, with a factorized density in the three angles. Our construction provides a concrete tool suited for applications of non-Archimedean models where an explicit angular description of invariant integration is required.