Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups

We provide a general expression of the Haar measure—that is, the essentially unique translation-invariant measure—on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the p-adic special orthogonal groups in dimension two, three and four (for every prime number p). In particular, the Haar measure on \(\text {SO}(2,\mathbb {Q}_p)\) $\text{SO}(2,{\mathbb{Q}}_p)$ is obtained by a direct application of our general formula. As for \(\text {SO}(3,\mathbb {Q}_p)\) $\text{SO}(3,{\mathbb{Q}}_p)$ and \(\text {SO}(4,\mathbb {Q}_p)\) $\text{SO}(4,{\mathbb{Q}}_p)$ , instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain p-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field \(\mathbb {Q}_p\) ${\mathbb{Q}}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the p-adic special orthogonal groups, with potential applications in p-adic quantum mechanics and in the recently proposed p-adic quantum information theory.